= Solution
Expanding the two products of <Householder reflections> gives
$$
\boxed{BA-AB=4(\mathbf a\cdot\mathbf b)(\mathbf b\mathbf a^T-\mathbf a\mathbf b^T).}
$$
The <commuting plane reflections> criterion follows: the operator vanishes if the normals are <orthogonal>, or if they are parallel. Conversely, if $c=\mathbf a\cdot\mathbf b\ne0$ 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 \b[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.
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