 Weyl group of a disconnected compact Lie group

For a compact Lie group that may be disconnected, fix a maximal torus $T$ and define the normalizer quotient $N_G(T)/T$. It is finite. To prove this, identify $T$ with its Lie algebra modulo the kernel lattice of its exponential map. A torus automorphism preserves this lattice. Its differential is therefore an integral invertible matrix in a lattice basis. A connected subgroup of the normalizer induces a continuous family of such matrices, hence acts trivially on $T$.

The identity component $N_G(T{)}^0$ thus centralizes $T$ and lies in $G^0$. The maximal-torus centralizer theorem for the connected compact group gives $N_G(T{)}^0=T$. The quotient is consequently discrete and compact, hence finite. Its action on $T$ need not be faithful when $G$ is disconnected: for a finite group, $T$ is trivial and $N_G(T)/T=G$. This is why the quotient $N_G(T)/C_G(T)$ must not silently replace $N_G(T)/T$ in this convention.