I am wondering how Lie's theorem and Engel's theorem fit into the theory of Lie algebras naturally, perhaps they are motivated by the Levi decomposition and the JNF decomposition?
I find it jarring to prove Engel for nilpotent Lie algebras for no real reason, but nilpotent operators arise in the JNF naturally, so perhaps this motivates Engel's theorem? It would be great to run into the necessity of needing these theorems without realizing it, as can be done for the Levi decomposition!
Overall, my best attempt so far is:
In a general real/complex Lie algebra $L$, if you write a JNF decomposition $$x = n + s,$$ with $s$ diagonalizable, $n$ nilpotent, what happens to $x$ when you do a Levi decomposition $$L = N \rtimes S,$$ $N$ solvable, $S$ semi-simple? On the one hand it kind of looks like a Levi decomposition is motivated by the JNF decomposition, but Erdmann's Lie Algebras book seems to only apply the JNF to the semi-simple part $S$ of $J$, so it seems like they are different things?
If the JNF applies only to the semi-simple part, then am I right in saying, given $L$, you first do a Levi decomposition $$L = N \rtimes S,$$ apply Lie's theorem on solvable Lie algebras to $N$ to decompose some of $x$ into upper triangular form $u$ and the rest, $x'$, then just needs to be dealt with, $$x = u + x',$$ so for $x'$ we can decompose it using the JNF into $$x' = n + s,$$ so that $$x = u + n + s,$$ and then on the nilpotent part $n$ we apply Engel's theorem to bring $n$ into strictly-upper-triangular form?