- To use OrthogonalComplement, you first need to load the SYM Package using Needs["SYM`"].

- The Lie algebra
*L*can be provided either from their structure constants or their infinitesimal generator representation. When providing the Lie algebra*L*with its structure constants denote the elements of the subspaces by using the symbol X, e.g. {X[1], X[4]}.

- If VL is a subspace of
*L,*then the set {xLk_{L}(x, y)=0 yV}*orthogonal complement*of*V*in*L*with respect to k_{L}.

- Depending of the input, OrthogonalComplement will return the base which spans the product space either by using the representation provided or the syntax for the subspaces using the symbol X.

- OrthogonalComplement[
*L, V, ip*] applies the function*ip*to pairs of elements to determine their orthogonality. The default function*ip*is KillingForm.