Remark 2

It follows from Cor. 5 that a maximal torus of G is a maximal commutative subgroup. The converse is not true: for example, in the group $\mathbf{S}\mathbf{O}(3,\mathbf{R})$, the maximal tori are of dimension 1, and thus cannot contain the subgroup of diagonal matrices, which is isomorphic to $(\mathbf{Z}/2\mathbf{Z})^2$. Moreover, if $g\in \mathbf{S}\mathbf{O}(3,\mathbf{R})$ is a non-scalar diagonal matrix, $g$ is a regular element of $\mathbf{S}\mathbf{O}(3,\mathbf{R})$ whose centralizer is not connected (cf. Cor. 4).