= Determinant-twist projection in odd dimension
{title2=$\pi(A)=(\det A)A$}
For real <orthogonal matrices> in odd dimension $d$, multiplying by their <determinant> defines a <group homomorphism> from $O(d)$ onto $SO(d)$. Scalar factors commute, so $\pi(AB)=\pi(A)\pi(B)$, while $\det\pi(A)=(\det A)^{d+1}=1$. The map fixes $SO(d)$ and has <kernel> $\{I,-I\}$. Since $-I$ is central and has determinant $-1$ in odd dimension, $O(d)\cong SO(d)\times C_2$. In even dimension the same expression does not map every orthogonal matrix into the <special orthogonal group>.
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