Exercise 8, § 5

Take $G =\mathbf{S}\mathbf{O}(8,\mathbf{R})/\{\pm I_8\}$; denote by $(\alpha_i)_{1\leq i\leq 4}$ a basis of $R(G,T)$ such that $\alpha_1, \alpha_3$ and $\alpha_4$ are not orthogonal to $\alpha_2$ (Chap. VI, Plate IV). Let A be the subgroup of T consisting of the $t\in T$ such that

Prove that A is isomorphic to $(\mathbf{Z}/2\mathbf{Z})^2$ and that its centralizer in G is a finite non-commutative group.