CHAPTER IX Compact Real Lie Groups

In this chapter$^1$, the expression “Lie group” means “finite dimensional Lie group over the field of real numbers”, the expression “Lie algebra” means, unless stated otherwise, “finite dimensional Lie algebra over the field of real numbers”, the expression “real Lie algebra” (resp. “complex Lie algebra”) means “finite dimensional Lie algebra over the field of real numbers (resp. “finite dimensional Lie algebra over the field of complex numbers”).

We denote by $G_0$ the identity component of a topological group G. We denote by C(G) the centre of a group G, by D(G) its derived group, and by $N_G(H)$ or N(H) (resp. $Z_G(H)$ or Z(H)) the normalizer (resp. centralizer) of a subset H of a group G.