Lie ideal

Remarks

Since $[a, x] = -[x, a]$, there is no difference between left and right ideals. Every ideal is a Lie subalgebra. If $\mathfrak a$ is an ideal, the quotient space $\mathfrak g / \mathfrak a$ is a Lie algebra with $[x + \mathfrak a, y + \mathfrak a] = [x, y] + \mathfrak a$.

Example: $\mathfrak{sl}_n(k)$, the trace-zero matrices, is an ideal of $\mathfrak{gl}_n(k)$. For all $X, Y$ one has $\tr(XY) = \tr(YX)$, so $\tr [X, Y] = 0$, and $[X, Y] \in \mathfrak{sl}_n(k)$.

Non-example: in $\mathfrak{sl}_2(k)$ with $\operatorname{char} k \ne 2$, take $h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ and $e = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$. The line $k h$ is a subalgebra but not an ideal, because $[e, h] = -2e$ is not a multiple of $h$.