Commuting plane reflections ($H_b{H}_a-H_a{H}_b=4(a\cdot b)(b{a}^T-a{b}^T)$)

For real unit vectors $a,b$, the Householder reflections $H_a=I-2a{a}^T$ and $H_b=I-2b{b}^T$ commute exactly when their normals are parallel or orthogonal. Expanding their products proves the formula. If $a\cdot b\ne 0$, a zero commutator requires $b{a}^T=a{b}^T$; applying this identity to $a$ gives $b=(a\cdot b)a$, hence parallelism. Orthogonal normals make the prefactor zero.

Equality on one selected vector is weaker than equality of the operators. For nonparallel, nonorthogonal normals in three dimensions, the commutator's kernel is the line perpendicular to both normals, spanned by $a\times b$. A vector on that line is fixed by both reflections even though the operators do not commute.