 Solution

Expanding the two products of Householder reflections givesThe commuting plane reflections criterion follows: the operator vanishes if the normals are orthogonal, or if they are parallel. Conversely, if $c=\mathbf{a}\cdot \mathbf{b}\ne 0$ and the operator vanishes, applying $\mathbf{b}{\mathbf{a}}^T-\mathbf{a}{\mathbf{b}}^T$ to $\mathbf{a}$ gives $\mathbf{b}-c\mathbf{a}=0$, proving parallelism.

This establishes equality of the reflected directions for every incident ray exactly in those two cases. For one fixed ray the claim needs that qualification: if nonparallel normals are not orthogonal, a ray along $\mathbf{a}\times \mathbf{b}$ has zero normal component for both mirrors and is unchanged in either order. Algebraically it lies in the kernel of the commutator; it is a special direction rather than a pair of commuting reflection operators.