## ABSTRACT

The present chapter is devoted to a very thorough investigation of compact Lie groups and culminates in their complete classification. The first point in the program is to obtain a necessary and sufficient condition in order that a given real Lie algebra R should be the Lie algebra of a compact Lie group. It turns out that the desired criterion consists in the existence on the vector space of the Lie algebra of a scalar product (u, v) which turns it into a Euclidean space and is invariant with respect to the adjoint group (see Section 54, F)). Invariance with respect to the adjoint group amounts to the condition