$\operatorname{SO}(3)$ dimensionality
===

- $R^TR = I$
- $\operatorname{det}R = 1$

I.e.
$\langle R_{[i,:]}, R_{[:,j]}\rangle = \delta_{ij},~i,j\in\{1..3\}^2$, total of $\frac12 3\cdot4=6$ unique constraints for $9$ variables

The $\operatorname{det}R=1$ constraint is redundant then