$\mathrm{Define}\left(A\,B\,C\,F\,G\right)\:$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\mathrm{EQ}≔A\left[\mathrm{\mu }\right]=G\left[\mathrm{\nu },\mathrm{\alpha }\right]A\left[\mathrm{\alpha }\right]F\left[\mathrm{\mu },\mathrm{\nu }\right]$

$\mathrm{EQ}≔A_{\mathrm{\mu }}=G_{\mathrm{\nu },\mathrm{\alpha }}{A}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{F}_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}$

The repeated and free indices of the lhs and rhs of this substitution equation eq

$\mathrm{Check}\left(\mathrm{EQ}\,\mathrm{all}\right)$

$\mathrm{The\; repeated\; indices\; per\; term\; are:}\left[\left\{\mathrm{...}\right\}\,\left\{\mathrm{...}\right\}\,\mathrm{...}\right]\mathrm{,\; the\; free\; indices\; are:}\left\{\mathrm{...}\right\}$

$\left(\left[\varnothing \right]\,\left\{\mathrm{\mu }\right\}\right)=\left(\left[\left\{\mathrm{\alpha }\,\mathrm{\nu }\right\}\right]\,\left\{\mathrm{\mu }\right\}\right)$

The easy case

$\mathrm{Substitute}\left(\mathrm{EQ}\,A\left[\mathrm{\mu }\right]\right)$

$G_{\mathrm{\nu },\mathrm{\alpha }}{A}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{F}_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}$

Now the free index in the target expression is now not $\mathrm{mu}$ but $\mathrm{nu}$

$\mathrm{Substitute}\left(\mathrm{EQ}\,A\left[\mathrm{\nu }\right]\right)$

$G_{\mathrm{\beta },\mathrm{\alpha }}{A}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{F}_{\mathrm{\nu }\phantom{\mathrm{\beta }}}^{\phantom{\mathrm{\nu }}\mathrm{\beta }}$

Distinction between covariant and contravariant indices

$\mathrm{Substitute}\left(\mathrm{EQ}\,A\left[\mathrm{~nu}\right]\right)$

$G_{\mathrm{\beta },\mathrm{\alpha }}{A}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{F}_{\phantom{}\phantom{\mathrm{\nu },\mathrm{\beta }}}^{\phantom{}\mathrm{\nu },\mathrm{\beta }}$

The index $\mathrm{nu}$ found repeated in the rhs of the substitution equation eq also appears repeated in the following target expression

$\mathrm{ee}≔A\left[\mathrm{\nu }\right]A\left[\mathrm{\nu }\right]$

$\mathrm{ee}≔A_{\mathrm{\nu }}{A}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}$

$\mathrm{res}≔\mathrm{Substitute}\left(\mathrm{EQ}\,\mathrm{ee}\right)$

$\mathrm{res}≔G_{\mathrm{\lambda },\mathrm{\kappa }}{A}_{\phantom{}\phantom{\mathrm{\kappa }}}^{\phantom{}\mathrm{\kappa }}{F}_{\mathrm{\nu }\phantom{\mathrm{\lambda }}}^{\phantom{\mathrm{\nu }}\mathrm{\lambda }}{G}_{\mathrm{\beta },\mathrm{\alpha }}{A}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{F}_{\phantom{}\phantom{\mathrm{\nu },\mathrm{\beta }}}^{\phantom{}\mathrm{\nu },\mathrm{\beta }}$

The repeated and free indices of this result

$\mathrm{Check}\left(\mathrm{res}\,\mathrm{all}\right)$

$\mathrm{The\; repeated\; indices\; per\; term\; are:}\left[\left\{\mathrm{...}\right\}\,\left\{\mathrm{...}\right\}\,\mathrm{...}\right]\mathrm{,\; the\; free\; indices\; are:}\left\{\mathrm{...}\right\}$

$\left[\left\{\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\kappa }\,\mathrm{\lambda }\,\mathrm{\nu }\right\}\right],\varnothing$

Functionality is also taken as parameters (only when the lhs is a Tensor function)

$\mathrm{Substitute}\left(A\left[\mathrm{\mu }\right]\left(X\right)=B\left[\mathrm{\mu }\right]\left(X\right)\,A\left[\mathrm{\nu }\right]\left(Y\right)\right)$

$B_{\mathrm{\nu }}\left(Y\right)$

Substituting in sub-expressions like algsubs

$\mathrm{EQ2}≔A\left[\mathrm{\mu }\right]\left(X\right)B\left[\mathrm{\nu }\right]\left(Y\right)=G\left[\mathrm{\mu },\mathrm{\nu }\right]\left(X\right)$

$\mathrm{EQ2}≔A_{\mathrm{\mu }}\left(X\right)B_{\mathrm{\nu }}\left(Y\right)=G_{\mathrm{\mu },\mathrm{\nu }}\left(X\right)$

$\mathrm{ee2}≔A\left[\mathrm{\alpha }\right]\left(X\right)B\left[\mathrm{\beta }\right]\left(Y\right)A\left[\mathrm{\rho }\right]\left(X\right)B\left[\mathrm{\rho }\right]\left(Y\right)$

$\mathrm{ee2}≔A_{\mathrm{\alpha }}\left(X\right)B_{\mathrm{\beta }}\left(Y\right)A_{\mathrm{\rho }}\left(X\right)B_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(Y\right)$

When substituting EQ2 into ee2, the sub-expression A . B appears twice, and not in the indets (indeterminates) of ee2:

$\mathrm{indets}\left(\mathrm{ee2}\right)$

$\left\{X\,Y\,A_{\mathrm{\alpha }}\left(X\right)\,A_{\mathrm{\rho }}\left(X\right)\,B_{\mathrm{\beta }}\left(Y\right)\,B_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(Y\right)\right\}$

$\mathrm{Substitute}\left(\mathrm{EQ2}\,\mathrm{ee2}\right)$

$G_{\mathrm{\alpha },\mathrm{\beta }}\left(X\right)G_{\mathrm{\rho }\phantom{\mathrm{\rho }}}^{\phantom{\mathrm{\rho }}\mathrm{\rho }}\left(X\right)$

Set a kind of letter to represent spinorindices and define a tensor with mixed spacetime and spinor indices

$\mathrm{Setup}\left(\mathrm{spinorindices}=\mathrm{lowercaselatin\_is}\right)$

$\left[\mathrm{spinorindices}=\mathrm{lowercaselatin\_is}\right]$

$\mathrm{Define}\left(T\left[\mathrm{\mu },i,j\right]\right)$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{B\,C\,A_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,F_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}\,G_{\mathrm{\nu },\mathrm{\alpha }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,T_{\mathrm{\mu },i,j}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

Substitutions in doubly indexed Dirac matrices

$\mathrm{Substitute}\left(\mathrm{Dgamma}\left[\mathrm{\mu }\right]\left[i,j\right]=T\left[\mathrm{\mu },i,j\right]\,\mathrm{Dgamma}\left[\mathrm{~alpha}\right]\left[\mathrm{~k},m\right]\right)$

$T_{\phantom{}\phantom{\mathrm{\alpha },k}m}^{\phantom{}\mathrm{\alpha },k\phantom{m}}$

When the free indices of left and right hand sides of a substitution equation are different, the substitution is halted and a related error message is displayed

$\mathrm{Substitute}\left(\mathrm{Dgamma}\left[\mathrm{\mu }\right]\left[i,j\right]=1\,\mathrm{Dgamma}\left[\mathrm{~alpha}\right]\left[\mathrm{~k},m\right]\right)$

$\mathrm{Free\; indices\; on\; both\; sides\; of\; the\; equation\; are\; different:\; found}\left\{i\,j\,\mathrm{\mu }\right\}\mathrm{on\; the\; left-hand\; side\; and}\varnothing \mathrm{on\; the\; right-hand\; side}$

Error, (in Physics:-SubstituteTensor) free indices on both sides of the equation are different

Substitutions of that kind could however be intentional; in those situations you can use the option disregardfreeindices

$\mathrm{Substitute}\left(\mathrm{Dgamma}\left[\mathrm{\mu }\right]\left[i,j\right]=1\,\mathrm{Dgamma}\left[\mathrm{~alpha}\right]\left[\mathrm{~k},m\right]\,\mathrm{disregardfreeindices}\right)$

$1$

Define a couple of arbitrary spacetime tensors for exploration purposes

$\mathrm{Define}\left(A\,B\right)$

$\mathrm{Defined\; as\; tensors}$

$\left\{B\,C\,A_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,F_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}\,G_{\mathrm{\nu },\mathrm{\alpha }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,T_{\mathrm{\mu },i,j}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

Enter, for example, this tensorial expression, with a contravariant free index $\mathrm{\rho }$

$\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]A\left[\mathrm{~mu}\right]\mathrm{g\_}\left[\mathrm{~alpha},\mathrm{~nu}\right]B\left[\mathrm{\nu },\mathrm{\sigma },\mathrm{~rho}\right]$

$g_{\mathrm{\alpha },\mathrm{\mu }}{A}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{g}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\nu }}{B}_{\mathrm{\nu },\mathrm{\sigma }\phantom{\mathrm{\rho }}}^{\phantom{\mathrm{\nu }}\phantom{,\mathrm{\sigma }}\mathrm{\rho }}$

To check the repeated and free indices in an expression use Check

$\mathrm{Check}\left(\,\mathrm{all}\right)$

$\mathrm{The\; repeated\; indices\; per\; term\; are:}\left[\left\{\mathrm{...}\right\}\,\left\{\mathrm{...}\right\}\,\mathrm{...}\right]\mathrm{,\; the\; free\; indices\; are:}\left\{\mathrm{...}\right\}$

$\left[\left\{\mathrm{\alpha }\,\mathrm{\mu }\,\mathrm{\nu }\right\}\right],\left\{\mathrm{\sigma }\,\mathrm{~\ρ}\right\}$

So (3) has $\mathrm{\alpha }$ and the contravariant $\mathrm{\rho }$ as free indices. Substitute now $\mathrm{\alpha }=\mathrm{\beta }$: the standard subs command will only substitute the covariant $\mathrm{\alpha }$

$\mathrm{subs}\left(\mathrm{\alpha }=\mathrm{\beta }\,\right)$

$g_{\mathrm{\beta },\mathrm{\mu }}{A}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{g}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\nu }}{B}_{\mathrm{\nu },\mathrm{\sigma }\phantom{\mathrm{\rho }}}^{\phantom{\mathrm{\nu }}\phantom{,\mathrm{\sigma }}\mathrm{\rho }}$

Consequently, the resulting expression is not equivalent to (3): it now has four free indices

$\mathrm{Check}\left(\,\mathrm{all}\right)$

$\mathrm{The\; repeated\; indices\; per\; term\; are:}\left[\left\{\mathrm{...}\right\}\,\left\{\mathrm{...}\right\}\,\mathrm{...}\right]\mathrm{,\; the\; free\; indices\; are:}\left\{\mathrm{...}\right\}$

$\left[\left\{\mathrm{\mu }\,\mathrm{\nu }\right\}\right],\left\{\mathrm{\beta }\,\mathrm{\sigma }\,\mathrm{~\α}\,\mathrm{~\ρ}\right\}$

To substitute both covariant and contravariant repeated indices obtaining an expression equivalent to original one use

$\mathrm{Substitute}\left(\mathrm{\alpha }=\mathrm{\beta }\,\right)$

$g_{\mathrm{\beta },\mathrm{\mu }}{A}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{g}_{\phantom{}\phantom{\mathrm{\beta },\mathrm{\nu }}}^{\phantom{}\mathrm{\beta },\mathrm{\nu }}{B}_{\mathrm{\nu },\mathrm{\sigma }\phantom{\mathrm{\rho }}}^{\phantom{\mathrm{\nu }}\phantom{,\mathrm{\sigma }}\mathrm{\rho }}$

$\mathrm{Check}\left(\,\mathrm{all}\right)$

$\mathrm{The\; repeated\; indices\; per\; term\; are:}\left[\left\{\mathrm{...}\right\}\,\left\{\mathrm{...}\right\}\,\mathrm{...}\right]\mathrm{,\; the\; free\; indices\; are:}\left\{\mathrm{...}\right\}$

$\left[\left\{\mathrm{\beta }\,\mathrm{\mu }\,\mathrm{\nu }\right\}\right],\left\{\mathrm{\sigma }\,\mathrm{~\ρ}\right\}$

Substitute now the contravariant $\mathrm{\rho }$ by $\mathrm{\tau }$; the standard subs command will fail because contravariant indices are prefixed by ~, so this returns (3) as given

$\mathrm{subs}\left(\mathrm{\rho }=\mathrm{\beta }\,\right)$

$g_{\mathrm{\alpha },\mathrm{\mu }}{A}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{g}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\nu }}{B}_{\mathrm{\nu },\mathrm{\sigma }\phantom{\mathrm{\rho }}}^{\phantom{\mathrm{\nu }}\phantom{,\mathrm{\sigma }}\mathrm{\rho }}$

To make it work with subs you would need to substitute ~rho = ~beta instead. Using Substitute you get the desired result regardless of this subtlety

$\mathrm{Substitute}\left(\mathrm{\rho }=\mathrm{\beta }\,\right)$

$g_{\mathrm{\alpha },\mathrm{\mu }}{A}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{g}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\nu }}{B}_{\mathrm{\nu },\mathrm{\sigma }\phantom{\mathrm{\beta }}}^{\phantom{\mathrm{\nu }}\phantom{,\mathrm{\sigma }}\mathrm{\beta }}$

$\mathrm{Check}\left(\,\mathrm{all}\right)$

$\mathrm{The\; repeated\; indices\; per\; term\; are:}\left[\left\{\mathrm{...}\right\}\,\left\{\mathrm{...}\right\}\,\mathrm{...}\right]\mathrm{,\; the\; free\; indices\; are:}\left\{\mathrm{...}\right\}$

$\left[\left\{\mathrm{\alpha }\,\mathrm{\mu }\,\mathrm{\nu }\right\}\right],\left\{\mathrm{\sigma }\,\mathrm{~\β}\right\}$

To substitute several indices at once, enclosing or not the substitution equations in a set or a list

$\mathrm{Substitute}\left(\mathrm{\rho }=\mathrm{\beta }\,\mathrm{\alpha }=\mathrm{\gamma }\,\right)$

$g_{\mathrm{\gamma },\mathrm{\mu }}{A}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{g}_{\phantom{}\phantom{\mathrm{\gamma },\mathrm{\nu }}}^{\phantom{}\mathrm{\gamma },\mathrm{\nu }}{B}_{\mathrm{\nu },\mathrm{\sigma }\phantom{\mathrm{\beta }}}^{\phantom{\mathrm{\nu }}\phantom{,\mathrm{\sigma }}\mathrm{\beta }}$

After substituting, tensors are re-evaluated; in this example, $\mathrm{\nu }$ becomes 1 and the normalized form of g_ is presented with it in the first place, so switching places with $\mathrm{\mu }$

$\mathrm{g\_}\left[\mathrm{\mu },\mathrm{\nu }\right]$

$g_{\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{Substitute}\left(\mathrm{\nu }=1\,\right)$

$g_{\mathrm{\mu },1}$

$\mathrm{Substitute}\left(\mathrm{\nu }=1\,\mathrm{\mu }=1\,\right)$

$g_{1,1}$

Note the result above is not fully evaluated; in that sense, Substitute, the same as SubstituteTensorIndices, follows the design of subs. To have results fully evaluated, pass the keyword evaluateexpression

$\mathrm{Substitute}\left(\mathrm{\nu }=1\,\mathrm{\mu }=1\,\,\mathrm{evaluateexpression}\right)$

$\mathrm{-1}$