The derived series of a Lie algebra is the sequence of ideals D^k(g) in g defined inductively by D^0(g) = g and D^(k + 1)(g) = [D^k(g), D^k(g)].  Note that D^(k + 1)(g) is contained in D^k(g).

The lower central series of a Lie algebra is a sequence of ideals L^k(g) in g defined inductively by L^0(g) = g and L^(k + 1)(g) = [g, L^k(g)].  Note that L^(k + 1)(g) is contained in L^k(g).

The upper central series of a Lie algebra is a sequence of ideals C^k(g) in g defined inductively by C^0(g) = GeneralizedCenter(0) and C^(k + 1)(g) = GeneralizedCenter(C^k(g)).  Note that C^(k)(g) is contained in C^(k + 1)(g).  If h is an ideal of the Lie algebra g, then GeneralizedCenter(h) is the ideal of vectors x in g such that [x,y] in h for all y in g.

LieAlgebraSeries(AlgName, keyword) calculates the series defined by the keyword for the Lie algebra AlgName.  If the first argument AlgName is omitted, then the appropriate series of the current Lie algebra is found.

LieAlgebraSeries(S, keyword) calculates the series defined by the keyword for the Lie subalgebra S (viewed as a Lie algebra in its own right).

LieAlgebraSeries returns a list of list of vectors L = [A_1, A_2, ...] where A_i is a basis for the (i - 1) term in the appropriate series.  The list L with A_m terminates if [i] A_(m - 1) = A_m; or [ii] in case of the derived and lower series if A_m = []; or [iii] in the case of the upper series A_1 = [] or A_m = all of g.

The dimensions of the subalgebras in these series can be easily computed with the Maple map and nops commands.

The command Series is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form Series(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Series(...).