The Structure of Compact Lie Groups
This chapter presents 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. A basic tool in the entire investigation is the introduction, on an arbitrary Lie algebra R, of a uniquely determined scalar product. A Lie algebra isomorphic with the Lie algebra of a compact Lie group will, in the sequel, be said to be compact. The bracket operation in a Lie algebra has certain features reminiscent of the familiar operation of vector multiplication. It turns out that every Lie algebra also admits, in a natural way, a scalar product, that is, a symmetric bilinear form. A Lie algebra is compact if and only if it admits a positive definite quadratic form which is invariant with respect to the adjoint group.