On the Lie Algebra Associated to the Canonical Matrices

For each fixed complex matrix M, the solution space to the matrix equation XM + MX^T = 0 is a Lie algebra denoted by \(\mathfrak{g}(n,M,\mathbb{C})\) $$\mathfrak{g}(n,M,\mathbb{C})$$ . We study the basic structure of \(\mathfrak{g}(n,M,\mathbb{C})\) $$\mathfrak{g}(n,M,\mathbb{C})$$ when M are the canonical matrices for congruence, i.e. M = J_n(0), Γ_n,H_2n(λ). We show that \(\mathfrak{g}(n,J_{n}(0),\mathbb{C})\) $$\mathfrak{g}(n,J_n(0),\mathbb{C})$$ (n even) and \(\mathfrak{g}(n,\Gamma_{n},\mathbb{C})\) $$\mathfrak{g}(n,{\mathrm{\Gamma }}_n,\mathbb{C})$$ are abelian Lie algebras; and \(\mathfrak{g}(n,J_{n}(0),\mathbb{C})\) $$\mathfrak{g}(n,J_n(0),\mathbb{C})$$ (n > 1 odd) is a non-nilpotent solvable Lie algebra. For \(\mathfrak{g}(2n,H_{2n}(\lambda),\mathbb{C})\) $$\mathfrak{g}(2n,H_{2n}(\lambda ),\mathbb{C})$$ , we determine its basis, radical and Levi subalgebra, and show that its radical is nilpotent.