Lie algebras of skew-adjoint endomorphisms of a bilinear form
lie algebra, skew-adjoint, bilinear form
M, equipped with a bilinear form, the skew-adjoint endomorphisms form a
Lie subalgebra of the Lie algebra of endomorphisms.
An equivalence of modules with bilinear forms gives equivalence of Lie algebras of skew-adjoint endomorphisms.
The Lie subalgebra of skew-adjoint square matrices corresponding to a square matrix
An invertible matrix
P gives a Lie algebra equivalence between those endomorphisms that are
skew-adjoint with respect to a square matrix
J and those with respect to
An equivalence of matrix algebras commuting with the transpose endomorphisms restricts to an equivalence of Lie algebras of skew-adjoint matrices.