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With the convention applicable also to disconnected compact groups, the Weyl group is $W=N_G(T)/T$, where the normalizer consists of elements taking $T$ to itself by conjugation. Put $\mathfrak{t}=L(T)$ and $\Lambda =\ker (\exp :\mathfrak{t}\to T)$. This is a full lattice with $T\cong \mathfrak{t}/\Lambda$.

If $\psi$ is a torus automorphism, exponential compatibility gives $\psi (\exp X)=\exp (D{\psi }_{I}X)$. Therefore $D{\psi }_I$ sends $\Lambda$ into itself; applying the inverse proves it sends the lattice onto itself. In a lattice basis this derivative is an element of $G{L}_r(\mathbb{Z})$.

Conjugation by a connected subgroup $Q\subset N_G(T)$ induces a continuous family of these integer matrices. Since $G{L}_r(\mathbb{Z})$ is discrete, this family is constant and equals its value at the identity. Its action on $\mathfrak{t}$ is therefore trivial, and surjectivity of the torus exponential shows that $Q$ centralizes $T$.

Apply this to the identity component $N_G(T{)}^0$. It lies in $G^0$ and centralizes $T$. The allowed maximal-abelian property of $T$ in $G^0$ gives $N_G(T{)}^0\subseteq T$, while the connected group $T$ lies in that identity component. Thus $N_G(T{)}^0=T$. The normalizer is a closed compact Lie subgroup, so its component quotient is discrete and compact, hence finite. Consequently $W$ is finite and discrete. For disconnected $G$, its action on the lattice need not be faithful; the component argument is essential.

The distinction is recorded in Weyl group of a disconnected compact Lie group.