Classical complex matrix Lie algebras

All matrix spaces below carry the commutator bracket $[A,B]=AB-BA$ and are Lie subalgebras of ${\mathfrak{g}\mathfrak{l}}_m(\mathbb{C})=M_m(\mathbb{C})$ in the sense of Lie algebras over a field; closure under the bracket is verified in each case by the computation displayed.

• The general linear Lie algebra ${\mathfrak{g}\mathfrak{l}}_n(\mathbb{C})=M_n(\mathbb{C})$, for $n\ge 1$. The special linear Lie algebra ${\mathfrak{s}\mathfrak{l}}_n(\mathbb{C})=\{A\in M_n(\mathbb{C}):\mathrm{tr}A=0\}$ is a Lie subalgebra because $\mathrm{tr}(AB-BA)=0$, and ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ agrees with The special linear Lie algebra sl_2.
• The symplectic Lie algebra ${\mathfrak{s}\mathfrak{p}}_{2n}(\mathbb{C})$ is the set of $A\in M_{2n}(\mathbb{C})$ with $AJ+J{A}^T=0$, for $n\ge 1$, where $J=\left(\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right)$.
• The orthogonal Lie algebras ${\mathfrak{s}\mathfrak{o}}_{2n}(\mathbb{C})$ and ${\mathfrak{s}\mathfrak{o}}_{2n+1}(\mathbb{C})$ are the sets of $A\in M_m(\mathbb{C})$ with $AJ+J{A}^T=0$, for $n\ge 1$, where $J=\left(\begin{array}{cc}0& I_n\\ I_n& 0\end{array}\right)$ for $m=2n$ and $J=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& I_n\\ 0& I_n& 0\end{array}\right)$ for $m=2n+1$.

Solving $AJ+J{A}^T=0$ blockwise gives $A=\left(\begin{array}{cc}a& b\\ c& -a^T\end{array}\right)$ for the symplectic and even orthogonal cases, with $b,c$ symmetric for ${\mathfrak{s}\mathfrak{p}}_{2n}(\mathbb{C})$ and $b,c$ skew-symmetric for ${\mathfrak{s}\mathfrak{o}}_{2n}(\mathbb{C})$, and gives $A=\left(\begin{array}{ccc}0& u& -w^T\\ w& a& b\\ -u^T& c& -a^T\end{array}\right),u\in M_{1\times n}(\mathbb{C}),w\in M_{n\times 1}(\mathbb{C}),b,c\text{ skew-symmetric},$ for ${\mathfrak{s}\mathfrak{o}}_{2n+1}(\mathbb{C})$. In particular $\dim {\mathfrak{s}\mathfrak{p}}_{2n}(\mathbb{C})=n(2n+1)=\dim {\mathfrak{s}\mathfrak{o}}_{2n+1}(\mathbb{C})$ and $\dim {\mathfrak{s}\mathfrak{o}}_{2n}(\mathbb{C})=n(2n-1)$. Each set is closed under the bracket: if $AJ=-J{A}^T$ and $BJ=-J{B}^T$, then $[A,B]J+J[A,B{]}^T=ABJ-BAJ+J{B}^T{A}^T-J{A}^T{B}^T$, and substituting $AJ=-J{A}^T$ and $BJ=-J{B}^T$ (equivalently $J{A}^T=-AJ$ and $J{B}^T=-BJ$) makes the four terms cancel in pairs. For $n\ge 1$, each symplectic or orthogonal family is a nonzero proper subspace of $M_m(\mathbb{C})$ closed under the commutator, hence a Lie subalgebra.