This came up doing some research in quantum information. Let us consider two orthogonal three-dimensional unit vectors $v$ and $w$
$v^T\cdot w=0$,
and the Householder transformation
$H=I_{3}-2v\cdot v^{T}$,
where $I_{3}$ is the $3\times3$ identity matrix. Then we have
$Hv=-v$,
$Hw=w$.
If we now multiply $H$ by $-I_3$, which is the three-dimensional reflection through the origin (parity transformation for physicists) we obtain an orthogonal matrix with unit determinant, hence a member of SO(3)
$\Omega\equiv-I_{3}H=2v\cdot v^{T}-I_3$.
But clearly now
$\Omega w=-w$
even though $\Omega\in\text{SO}(3)$. So my questions are: how is this possible? how come that an SO(3) transformation is able to produce a parity transformation? is this trivial or there's a deep mathematical reason?