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?