I am reading Humphreys' book "Introduction to Lie algebras and representation theory."
A Lie algebra $\mathfrak{g}$ is called nilpotent if $[\mathfrak{g},[\mathfrak{g},\cdots,[\mathfrak{g},\mathfrak{g}]]]]]=\{0\}$ for some number of repetitions. What are some enlightening examples of nilpotent Lie algebras? Are there any examples of nilpotent Lie algebras for which Engel's theorem is useful in establishing nilpotency?
I am aware that the Lie algebra of strictly upper triangular matrices is nilpotent, as seems very straightforward to verify (Humphreys cites this as a consequence of Engel's theorem but this seems like an overcomplication).
Wikipedia lists two extra examples:
A "Cartan subalgebra" is always nilpotent, but this appears to be part of the definition, so it seems like examples can only be as interesting as good examples of Lie algebras which arise as Cartan subalgebras. Are there any?
And (according to wikipedia) if a Lie algebra has an automorphism (I assume an automorphism of the Lie algebra and not of the vector space) of prime period and no nontrivial fixed points, then the Lie algebra is nilpotent. I have no idea how natural this is and I have no idea what examples of this would look like.
I would prefer examples which are real or complex Lie algebras but I would be interested in any answer.