RootSpaceDecomposition - Maple Help

LieAlgebras[RootSpaceDecomposition] - find the root space decomposition for a semi-simple Lie algebra from a Cartan subalgebra

Calling Sequences

RootSpaceDecomposition(CSA)

Parameters

CSA     - a list of vectors in a Lie algebra, defining a Cartan subalgebra

Description

 • Let g be a semi-simple Lie algebra and h a Cartan subalgebra. Let be a basis for. The linear transformations are simultaneously diagonalizable over C - if x ∈ g is a common eigenvector for all these transformations, then . The $m$-tuples  are called the roots of with respect to the Cartan sub-algebra and the root space decomposition of g with respect to h.
 • The roots and root space decomposition enjoy the following basic properties.
 1 The eigenspaces or root spaces are each 1-dimensional.
 2 Ifis a root, then so is
 3 If  and , then if is a root; otherwise
 4 If  and , then The vectors define a 3-dimensional Lie algebra isomorphic to $\mathrm{sl}\left(2\right)$.
 5 If  and and then where is the Killing form.
 6 The Killing form is non-degenerate on h.
 7 The number of linearly independent roots is $m.$
 • The command RootSpaceDecomposition returns a table describing the root space decomposition of g with respect to h. The indices of the table are the roots ${\mathrm{\alpha }}_{}$ and the table entries are vectors in g defining the root spaces 
 • The command Query/"RootSpaceDecomposition" will check that a given table defines a root space decomposition.
 • The commands SimpleLieAlgebraData and SimpleLieAlgebraProperties can be used to quickly obtain the root space decomposition for any simple classical matrix algebra.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

In this example we initialize the simple Lie algebra (of trace-free matrices), calculate a Cartan subalgebra and a root space decomposition. We then illustrate the above properties of the root space decomposition.

First, we use the program SimpleLieAlgebraData to generate the Lie algebra data for s$l\left(3\right)$.

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("sl\left(3\right)",\mathrm{sl3},\mathrm{labelformat}="gl",\mathrm{labels}=\left['E','\mathrm{\theta }'\right]\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e2}}\right]{,}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E21}}{,}{\mathrm{E23}}{,}{\mathrm{E31}}{,}{\mathrm{E32}}\right]{,}\left[{\mathrm{θ11}}{,}{\mathrm{θ22}}{,}{\mathrm{θ12}}{,}{\mathrm{θ13}}{,}{\mathrm{θ21}}{,}{\mathrm{θ23}}{,}{\mathrm{θ31}}{,}{\mathrm{θ32}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: sl3}}$ (2.2)

The program CartanSubalgebra calculates a Cartan subalgebra for $\mathrm{sl}\left(3\right)$.

 sl3 > $\mathrm{CSA}≔\mathrm{CartanSubalgebra}\left(\mathrm{sl3}\right)$
 ${\mathrm{CSA}}{:=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}\right]$ (2.3)

Now compute the root space decomposition. We see that each root space is 1-dimensional (Property 1).

 sl3 > $\mathrm{RSD}≔\mathrm{RootSpaceDecomposition}\left(\mathrm{CSA}\right)$
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{-}{2}{,}{-}{1}\right]{=}{\mathrm{E31}}{,}\left[{2}{,}{1}\right]{=}{\mathrm{E13}}{,}\left[{1}{,}{2}\right]{=}{\mathrm{E23}}{,}\left[{1}{,}{-}{1}\right]{=}{\mathrm{E12}}{,}\left[{-}{1}{,}{1}\right]{=}{\mathrm{E21}}{,}\left[{-}{1}{,}{-}{2}\right]{=}{\mathrm{E32}}\right]\right)$ (2.4)

The roots are the indices for this table, given as column vectors. It is easy to see that the negative of any root is a root (Property 2).

 sl3 > $\mathrm{RT}≔\left[⟨-1,-2⟩,⟨1,2⟩,⟨-1,1⟩,⟨2,1⟩,⟨-2,-1⟩,⟨1,-1⟩\right]$
 ${\mathrm{RT}}{:=}\left[\left[\begin{array}{r}{-}{1}\\ {-}{2}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {2}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{2}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{-}{2}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {-}{1}\end{array}\right]\right]$ (2.5)

Here are the eigenvectors or root spaces.

 sl3 > $\mathrm{RS}≔\left[\mathrm{seq}\left(\mathrm{RSD}\left[\mathrm{convert}\left(v,\mathrm{list}\right)\right],v=\mathrm{RT}\right)\right]$
 ${\mathrm{RS}}{:=}\left[{\mathrm{E32}}{,}{\mathrm{E23}}{,}{\mathrm{E21}}{,}{\mathrm{E13}}{,}{\mathrm{E31}}{,}{\mathrm{E12}}\right]$ (2.6)

The 2nd and 6th roots add to give the 4th root. This means that the Lie bracket of the 3rd and 4th vectors in (2.6) should be a multiple of the 1st vector (Property 3).

 sl3 > $\mathrm{LieBracket}\left(\mathrm{RS}\left[2\right],\mathrm{RS}\left[6\right]\right)$
 ${-}{\mathrm{E13}}$ (2.7)

The 1st and 2nd roots are negatives of each other so the Lie bracket of the 1st and 2nd vectors in (2.6)should belong to the Cartan subalgebra (Property 4).

 sl3 > $H≔\mathrm{LieBracket}\left(\mathrm{RS}\left[1\right],\mathrm{RS}\left[2\right]\right)$
 ${H}{:=}{-}{\mathrm{E22}}$ (2.8)

The vectors  form a 3-dimensional Lie algebra (Property 4).

 sl3 > $\mathrm{LieAlgebraData}\left(\left[H,\mathrm{RS}\left[1\right],\mathrm{RS}\left[2\right]\right]\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}\right]$ (2.9)

The Killing form restricted to the root spaces of the 2nd, 4rd and 6th roots is diagonal (Property 5).

 sl3 > $\mathrm{Killing}\left(\left[\mathrm{RS}\left[2\right],\mathrm{RS}\left[4\right],\mathrm{RS}\left[6\right]\right]\right)$
 $\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]$ (2.10)

Example 2.

We repeat the analysis of Example 1 using the Lie algebra This is a 21-dimensional Lie algebra of 7×7 matrices which preserve the quadratic form First, we use the program SimpleLieAlgebraData to generate the Lie algebra data for s$l\left(3\right)$.

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("so\left(4,3\right)",\mathrm{so43},\mathrm{labelformat}="gl",\mathrm{labels}=\left['R','\mathrm{\sigma }'\right]\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e16}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e19}}\right]{=}{-}{\mathrm{e19}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e17}}\right]{=}{\mathrm{e16}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e19}}\right]{=}{-}{\mathrm{e20}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e9}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e18}}\right]{=}{\mathrm{e16}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e19}}\right]{=}{-}{\mathrm{e21}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e17}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e20}}\right]{=}{-}{\mathrm{e19}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e17}}\right]{=}{\mathrm{e17}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e20}}\right]{=}{-}{\mathrm{e20}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e5}}{-}{\mathrm{e9}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e18}}\right]{=}{\mathrm{e17}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e20}}\right]{=}{-}{\mathrm{e21}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e18}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e19}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e17}}\right]{=}{\mathrm{e18}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e20}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e18}}\right]{=}{\mathrm{e18}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e21}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e1}}{-}{\mathrm{e5}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e19}}\right]{=}{\mathrm{e17}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e20}}\right]{=}{-}{\mathrm{e16}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e1}}{-}{\mathrm{e9}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e19}}\right]{=}{\mathrm{e18}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e16}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e5}}{-}{\mathrm{e9}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e20}}\right]{=}{\mathrm{e18}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e17}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e20}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e17}}\right]{=}{-}{\mathrm{e19}}{,}\left[{\mathrm{e14}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e21}}{,}\left[{\mathrm{e14}}{,}{\mathrm{e18}}\right]{=}{-}{\mathrm{e19}}{,}\left[{\mathrm{e15}}{,}{\mathrm{e17}}\right]{=}{\mathrm{e21}}{,}\left[{\mathrm{e15}}{,}{\mathrm{e18}}\right]{=}{-}{\mathrm{e20}}{,}\left[{\mathrm{e16}}{,}{\mathrm{e17}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e16}}{,}{\mathrm{e18}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e16}}{,}{\mathrm{e19}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e16}}{,}{\mathrm{e20}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e16}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e17}}{,}{\mathrm{e18}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e17}}{,}{\mathrm{e19}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e17}}{,}{\mathrm{e20}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e17}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e18}}{,}{\mathrm{e19}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e18}}{,}{\mathrm{e20}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e18}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e19}}{,}{\mathrm{e20}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e19}}{,}{\mathrm{e21}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e20}}{,}{\mathrm{e21}}\right]{=}{\mathrm{e15}}\right]{,}\left[{\mathrm{R11}}{,}{\mathrm{R12}}{,}{\mathrm{R13}}{,}{\mathrm{R21}}{,}{\mathrm{R22}}{,}{\mathrm{R23}}{,}{\mathrm{R31}}{,}{\mathrm{R32}}{,}{\mathrm{R33}}{,}{\mathrm{R15}}{,}{\mathrm{R16}}{,}{\mathrm{R26}}{,}{\mathrm{R42}}{,}{\mathrm{R43}}{,}{\mathrm{R53}}{,}{\mathrm{R17}}{,}{\mathrm{R27}}{,}{\mathrm{R37}}{,}{\mathrm{R47}}{,}{\mathrm{R57}}{,}{\mathrm{R67}}\right]{,}\left[{\mathrm{σ11}}{,}{\mathrm{σ12}}{,}{\mathrm{σ13}}{,}{\mathrm{σ21}}{,}{\mathrm{σ22}}{,}{\mathrm{σ23}}{,}{\mathrm{σ31}}{,}{\mathrm{σ32}}{,}{\mathrm{σ33}}{,}{\mathrm{σ15}}{,}{\mathrm{σ16}}{,}{\mathrm{σ26}}{,}{\mathrm{σ42}}{,}{\mathrm{σ43}}{,}{\mathrm{σ53}}{,}{\mathrm{σ17}}{,}{\mathrm{σ27}}{,}{\mathrm{σ37}}{,}{\mathrm{σ47}}{,}{\mathrm{σ57}}{,}{\mathrm{σ67}}\right]$ (2.11)
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: so43}}$ (2.12)

The program CartanSubalgebra calculates a Cartan subalgebra for $\mathrm{so}\left(4,3\right)$.

 so43 > $\mathrm{CSA}≔\mathrm{CartanSubalgebra}\left(\mathrm{so43}\right)$
 ${\mathrm{CSA}}{:=}\left[{\mathrm{R11}}{,}{\mathrm{R22}}{,}{\mathrm{R33}}\right]$ (2.13)

Now compute the root space decomposition. We see that each root space is 1-dimensional (Property 1).

 so43 > $\mathrm{RSD}≔\mathrm{RootSpaceDecomposition}\left(\mathrm{CSA}\right)$
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{1}{,}{0}{,}{0}\right]{=}{\mathrm{R17}}{,}\left[{-}{1}{,}{1}{,}{0}\right]{=}{\mathrm{R21}}{,}\left[{0}{,}{-}{1}{,}{0}\right]{=}{\mathrm{R57}}{,}\left[{0}{,}{1}{,}{0}\right]{=}{\mathrm{R27}}{,}\left[{0}{,}{-}{1}{,}{1}\right]{=}{\mathrm{R32}}{,}\left[{-}{1}{,}{0}{,}{1}\right]{=}{\mathrm{R31}}{,}\left[{0}{,}{0}{,}{1}\right]{=}{\mathrm{R37}}{,}\left[{0}{,}{0}{,}{-}{1}\right]{=}{\mathrm{R67}}{,}\left[{-}{1}{,}{-}{1}{,}{0}\right]{=}{\mathrm{R42}}{,}\left[{0}{,}{1}{,}{1}\right]{=}{\mathrm{R26}}{,}\left[{1}{,}{0}{,}{1}\right]{=}{\mathrm{R16}}{,}\left[{1}{,}{1}{,}{0}\right]{=}{\mathrm{R15}}{,}\left[{-}{1}{,}{0}{,}{0}\right]{=}{\mathrm{R47}}{,}\left[{0}{,}{1}{,}{-}{1}\right]{=}{\mathrm{R23}}{,}\left[{1}{,}{-}{1}{,}{0}\right]{=}{\mathrm{R12}}{,}\left[{-}{1}{,}{0}{,}{-}{1}\right]{=}{\mathrm{R43}}{,}\left[{1}{,}{0}{,}{-}{1}\right]{=}{\mathrm{R13}}{,}\left[{0}{,}{-}{1}{,}{-}{1}\right]{=}{\mathrm{R53}}\right]\right)$ (2.14)
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{-}{1}{,}{0}{,}{-}{1}\right]{=}{\mathrm{R43}}{,}\left[{0}{,}{1}{,}{0}\right]{=}{\mathrm{R27}}{,}\left[{0}{,}{1}{,}{-}{1}\right]{=}{\mathrm{R23}}{,}\left[{0}{,}{-}{1}{,}{0}\right]{=}{\mathrm{R57}}{,}\left[{-}{1}{,}{0}{,}{0}\right]{=}{\mathrm{R47}}{,}\left[{1}{,}{-}{1}{,}{0}\right]{=}{\mathrm{R12}}{,}\left[{0}{,}{1}{,}{1}\right]{=}{\mathrm{R26}}{,}\left[{1}{,}{1}{,}{0}\right]{=}{\mathrm{R15}}{,}\left[{0}{,}{-}{1}{,}{-}{1}\right]{=}{\mathrm{R53}}{,}\left[{-}{1}{,}{1}{,}{0}\right]{=}{\mathrm{R21}}{,}\left[{0}{,}{0}{,}{-}{1}\right]{=}{\mathrm{R67}}{,}\left[{-}{1}{,}{-}{1}{,}{0}\right]{=}{\mathrm{R42}}{,}\left[{0}{,}{-}{1}{,}{1}\right]{=}{\mathrm{R32}}{,}\left[{1}{,}{0}{,}{1}\right]{=}{\mathrm{R16}}{,}\left[{-}{1}{,}{0}{,}{1}\right]{=}{\mathrm{R31}}{,}\left[{1}{,}{0}{,}{0}\right]{=}{\mathrm{R17}}{,}\left[{0}{,}{0}{,}{1}\right]{=}{\mathrm{R37}}{,}\left[{1}{,}{0}{,}{-}{1}\right]{=}{\mathrm{R13}}\right]\right)$ (2.15)

The roots, given as column vectors, are obtained using LieAlgebraRoots. It is easy to see that the negative of any root is a root (Property 2).

 so43 > $\mathrm{RT}≔\mathrm{map}\left(\mathrm{Vector},\left[\left[0,1,-1\right],\left[1,0,0\right],\left[1,0,-1\right],\left[1,1,0\right],\left[0,-1,0\right],\left[0,0,-1\right],\left[0,1,1\right],\left[1,-1,0\right],\left[-1,0,-1\right],\left[-1,-1,0\right],\left[-1,0,0\right],\left[0,0,1\right],\left[1,0,1\right],\left[-1,0,1\right],\left[-1,1,0\right],\left[0,-1,1\right],\left[0,1,0\right],\left[0,-1,-1\right]\right]\right)$
 ${\mathrm{RT}}{:=}\left[\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {0}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{-}{1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {-}{1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {-}{1}\\ {-}{1}\end{array}\right]\right]$ (2.16)

Here are the eigenvectors or root spaces.

 so43 > $\mathrm{RS}≔\left[\mathrm{seq}\left(\mathrm{RSD}\left[\mathrm{convert}\left(v,\mathrm{list}\right)\right],v=\mathrm{RT}\right)\right]$
 ${\mathrm{RS}}{:=}\left[{\mathrm{R23}}{,}{\mathrm{R17}}{,}{\mathrm{R13}}{,}{\mathrm{R15}}{,}{\mathrm{R57}}{,}{\mathrm{R67}}{,}{\mathrm{R26}}{,}{\mathrm{R12}}{,}{\mathrm{R43}}{,}{\mathrm{R42}}{,}{\mathrm{R47}}{,}{\mathrm{R37}}{,}{\mathrm{R16}}{,}{\mathrm{R31}}{,}{\mathrm{R21}}{,}{\mathrm{R32}}{,}{\mathrm{R27}}{,}{\mathrm{R53}}\right]$ (2.17)

The 2nd and 12th roots add to give the 13th root. This means that the Lie bracket of the 1st and 7th vectors should be a multiple of the 10th vector (Property 3).

 so43 > $\mathrm{RT}\left[2\right]+\mathrm{RT}\left[12\right],\mathrm{RT}\left[13\right]$
 $\left[\begin{array}{r}{1}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\\ {1}\end{array}\right]$ (2.18)
 so43 > $\mathrm{LieBracket}\left(\mathrm{RS}\left[2\right],\mathrm{RS}\left[12\right]\right),\mathrm{RS}\left[13\right]$
 ${\mathrm{R16}}{,}{\mathrm{R16}}$ (2.19)

The Lie bracket of any root and its negative belongs to the Cartan subalgebra (Property 4).

 so43 > $H≔\mathrm{LieBracket}\left(\mathrm{RSD}\left[\left[1,0,1\right]\right],\mathrm{RSD}\left[-\left[1,0,1\right]\right]\right)$
 ${H}{:=}{-}{\mathrm{R11}}{-}{\mathrm{R33}}$ (2.20)

The Killing form restricted to the positive root space is diagonal (Property 5).

 so43 > $\mathrm{PR}≔\mathrm{PositiveRoots}\left(\mathrm{RT},⟨5,3,1⟩\right)$
 ${\mathrm{PR}}{:=}\left[\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {0}\end{array}\right]\right]$ (2.21)
 so43 > $\mathrm{PS}≔\left[\mathrm{seq}\left(\mathrm{RSD}\left[\mathrm{convert}\left(v,\mathrm{list}\right)\right],v=\mathrm{PR}\right)\right]$
 ${\mathrm{PS}}{:=}\left[{\mathrm{R23}}{,}{\mathrm{R17}}{,}{\mathrm{R13}}{,}{\mathrm{R15}}{,}{\mathrm{R26}}{,}{\mathrm{R12}}{,}{\mathrm{R37}}{,}{\mathrm{R16}}{,}{\mathrm{R27}}\right]$ (2.22)
 sl3 > $\mathrm{Killing}\left(\mathrm{PS}\right)$
 $\left[\begin{array}{rrrrrrrrr}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]$ (2.23)