= Proof of Engel theorem by induction and normalizers
First prove the <Engel lemma> by induction on the dimension of a <Lie algebra> of <nilpotent endomorphisms>. If $x^m=0$, then $\operatorname{ad}x=L_x-R_x$ on the endomorphism algebra has $(\operatorname{ad}x)^{2m-1}=0$, since left and right multiplication commute. Thus a proper <Lie subalgebra> $M$ acts nilpotently on $L/M$, and induction gives a nonzero class normalized by $M$. This proves the <Engel normalizer lemma>.
For a maximal proper $M$, its normalizer is all of $L$, so $M$ is an ideal. Maximality then forces $\dim L/M=1$, since every line in a Lie algebra is a subalgebra. Induction gives a nonzero common annihilator $W$ of $M$ in the representation space. Ideality makes $W$ stable under $L$. A generator $x$ of $L/M$ is nilpotent on $W$ and has nonzero kernel there, providing a vector killed by all of $L$. In the possibly nonfaithful quotient actions, apply induction to the image, whose dimension is at most $\dim M$.
Repeatedly apply this common-kernel statement to quotient representation spaces to obtain a complete flag with $xV_j\subseteq V_{j-1}$. Hence the operators are simultaneously strictly upper triangular. Applying this to the <Adjoint representation> shows that a finite-dimensional <Lie algebra> whose every adjoint map is nilpotent has terminating <lower central series>. Conversely, termination of that series immediately makes every adjoint map nilpotent. This proves <Engel theorem>.
// Target: semisimple-lie-algebra.bigb
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