User:Netrapt/sandbox

Examples
For $$\mathfrak{d}(2) \subset \mathfrak{gl}(2)$$, the commutator of two elements $$g \in \mathfrak{gl}(2)$$ and $$d \in \mathfrak{d}(2)$$: $$\begin{align} \left[ \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \begin{bmatrix} x & 0 \\ 0 & y \end{bmatrix} \right] &= \begin{bmatrix} ax & by\\ cx & dy \\ \end{bmatrix} - \begin{bmatrix} ax & bx\\ cy & dy \\ \end{bmatrix} \\ &= \begin{bmatrix} 0 & b(y-x) \\ c(x-y) & 0 \end{bmatrix}

\end{align}$$ shows $$\mathfrak{d}(2)$$ is a subalgebra, but not an ideal. In fact, every one-dimensional linear subspace of a Lie algebra has an induced abelian Lie algebra structure, which is generally not an ideal. For any simple Lie algebra, all abelian Lie algebras can never be ideals.

Netrapt (talk) 01:37, 5 November 2022 (UTC)