In my lecture notes I have stumbled upon the following:
"Indeed, the statement [Lie's theorem] immediately implies that any such representation [complex representation of a solvable Lie algebra] contains an invariant subspace of dimension one, which in particular implies that irreducible complex representations are automatically one-dimensional and trivial."
Two questions here:
Why is the prerequisite "solvable" no longer needed in the last statement? Or is it just true for any representation of a solvable Lie algebra?
I get that it must be one-dimensional (cf. for example Complex irreducible representation of solvable lie algebra). But why does it follow that such representation must be trivial?