Engel subalgebra

Remarks

The set $E_{\mathfrak g}(x)$ is the generalised eigenspace of $\operatorname{ad} x$ for the eigenvalue $0$. It is a subalgebra by the Leibniz rule $(\operatorname{ad} x)^n [y, z] = \sum_{i=0}^{n} \binom{n}{i} [(\operatorname{ad} x)^i y, (\operatorname{ad} x)^{n-i} z]$, and it contains $x$.

Example: in the special linear Lie algebra $\mathfrak{sl}_2(k)$ with $\operatorname{char} k \ne 2$ and basis $e, h, f$, the map $\operatorname{ad} h$ sends $e \mapsto 2e$, $h \mapsto 0$ and $f \mapsto -2f$. So $E(h) = kh$. Since $\operatorname{ad} e$ is nilpotent, with $(\operatorname{ad} e)^3 = 0$, one gets $E(e) = \mathfrak{sl}_2(k)$.

Non-example: $kh + ke$ is a subalgebra containing $h$, but it is not $E(h)$, since $(\operatorname{ad} h)^n e = 2^n e \ne 0$ for all $n$.

If $k$ is an algebraically closed field of characteristic $0$, the Engel subalgebras of smallest dimension are Cartan subalgebras.