# Advantages of solvability, nilpotency and semisimplicity of Lie algebras?

After pondering on the notion of solvability, nilpotency and semi-simplicity of linear Lie algebras for days (I have been reading Humphreys' Introduction to Lie algebra lately), I remember a professor saying this many many years ago, the special Euclidean group SE(3) is not interesting since it is neither solvable nor simple. I couldn't stop thinking about what it could mean.

As an engineer, I understand that by having semi-simplicity and compactness, the negative killing form serves as a bi-invariant Riemannian metric. Nilpotency of a Lie algebra leads to closed form solution to non-holonomic motion planning problems. If it is linear endomorphisms we are talking about, solvability, nilpotency corresponds to upper triangular matrices and strictly upper triangular matrices under appropriate basis. Semi-simple endomorphisms admit Jordan Chevalley decomposition. These are some potential advantages for engineering.

My question is, are there any general guidelines, either mathematical or engineering, to appreciate these structures?

The question is rather broad, and a short answer is difficult. If it comes to "appreciate these structres", one could argue that (finite-dimensional) complex semisimple Lie algebras can be nicely classified. The result and the methods are beautiful. Solvable and nilpotent Lie algebras have a much more complicated behaviour in this respect. However, they are equally important, e.g., arising in physics (Heisenberg Lie algebras are nilpotent) and many other areas (from geometry, number theory, etc). In general every finite-dimensional Lie algebra $L$ in characteristic zero has a Levi-decomposition $$L=S \rtimes rad(L)$$ with a semisimple Levi-subalgebra $S$ and the solvable radial $rad(L)$, which is the largest solvable ideal of $L$.