  Quantum $\mathcal{R}$-matrices are the building blocks for the colored HOMFLY
polynomials. In the case of three-strand braids with an identical
finite-dimensional irreducible representation $T$ of $SU_q(N)$ associated with
each strand one needs two matrices: $\mathcal{R}_1$ and $\mathcal{R}_2$. They
are related by the Racah matrices $\mathcal{R}_2 = \mathcal{U} \mathcal{R}_1
\mathcal{U}^{\dagger}$. Since we can always choose the basis so that
$\mathcal{R}_1$ is diagonal, the problem is reduced to evaluation of
$\mathcal{R}_2$-matrices. This paper is one more step on the road to
simplification of such calculations. We found out and proved for some cases
that $\mathcal{R}_2$-matrices could be transformed into a block-diagonal ones.
The essential condition is that there is a pair of accidentally coinciding
eigenvalues among eigenvalues of $\mathcal{R}_1$-matrix. The angle of the
rotation in the sectors corresponding to accidentally coinciding eigenvalues
from the basis defined by the Racah matrix to the basis in which
$\mathcal{R}_2$ is block-diagonal is $\pm \frac{\pi}{4}$.
