As we saw in Chap. 5 , properties of Lie groups are controlled to a considerable extent by their Lie algebras. So a study of Lie Groups is inevitably tied up with the study of Lie algebras. In this and the next chapter we study the structure of Lie algebras. This is done over an arbitrary field k of characteristic zero—the theorems we prove do not require the assumption that the ground field is a local field. The theorems have implications for Lie groups over k when k is a local field. We draw attention to some of these implications, but this is not done exhaustively. We prove theorems due to Engel and Lie which deal with nilpotent and solvable Lie algebras and a theorem of Cartan’s giving a criterion for the solvability of a Lie subalgebra of \(\mathfrak {gl}(V)\) .