= Cartan subalgebra existence from a minimal Engel subalgebra
{c}
For a finite-dimensional <Lie algebra> $L$ over an infinite <field>, let $L_{0,x}=\ker(\operatorname{ad}x)^{\dim L}$. A member of minimum dimension among these generalized zero spaces is a <Cartan subalgebra>, using the standard fact that each $L_{0,x}$ is a self-normalizing <Lie subalgebra>. The key minimality argument is more general: if $A\subseteq H=L_{0,x}$ and $H$ minimizes dimension for $x\in A$, then $H\subseteq L_{0,y}$ for every $y\in A$.
Indeed $H$ is preserved by $\operatorname{ad}(x+ty)$, and the induced operator on $L/H$ has determinant a polynomial in $t$ nonzero at zero, by the <finite-dimensional Fitting decomposition>. For all but finitely many $t$, its generalized zero space lies in $H$ and hence equals $H$ by minimality. The restriction to $H$ is therefore nilpotent for all such $t$. Its characteristic coefficients are polynomials in $t$ vanishing infinitely often; their highest-degree terms are the corresponding coefficients of $\operatorname{ad}y|_H$. These vanish, so that restriction is nilpotent. For a global minimum set $A=H$; <Engel's theorem> makes $H$ nilpotent and self-normalization gives the result.
Back to article page
