Using the Skolem-Noether theorem for finite-dimensional central simple algebras, we can quickly see that two linear operators over a finite dimensional vector space must be conjugated if they share a minimal polynomial. If the latter splits, this proves the existence of the Jordan normal form.
As a challenge, I attempt to solve exercises in Humphreys' Lie algebra book with as little space consumption as possible. This will force me to make the core argument rather clear and on-the-point.
The relation \(rRm \colon \Leftrightarrow rm=0\) between a commutative ring and a module over that ring gives rise to the notion of torsion sets, which must be submodules, and annihilator sets, which must be ideals. We provide a counterexample to the question whether every subset is the torsion set of something.
An important corollary of Engel's Theorem is that every \(\mathrm{ad}\)-nilpotent lie algebra must already be nilpotent. However, unpacking the proof of that requires some care.