 Solution

Use the given torus of independent rotations in $n$ mutually orthogonal two-planes and a fixed final axis. A normalizer element permutes the two-plane rotation characters, and on each plane either preserves or reverses its orientation. Its induced action therefore permutes the $n$ angle coordinates and changes their signs independently.

Every such signed permutation is realized. Permutations of two-dimensional blocks have determinant one. A reflection in any chosen plane reverses that angle; reflecting the remaining one-dimensional axis as necessary restores total determinant one. The kernel consists precisely of independent plane rotations: commuting with a generic torus element forces preservation of its distinct rotation planes, and the determinant condition then fixes the final axis. Thus the kernel is $T$ andThis is the signed permutation group, the Weyl group of type $B_n$.