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Rotations compatible with a lattice base system

If matrix A represents a type I symmetry operation, we can calculate the rotation angle $\alpha$ from the value of the A matrix trace. We must remember that the trace of A is invariant with respect to a base system transformation.

In a lattice base system the trace is an integer number, since the elements of the matrix are integers. In an orthonormal base system, the counter-clockwise rotation of an angle $\alpha$, for example, around the z axis is given by:

$$\left[\begin{array} {ccc} \cos \alpha& -\sin \alpha& 0\\ \sin \alpha& \cos \alpha& 0\\ 0&0&1\end{array}\right]$$

and then the trace is equal to $2 \cos \alpha + 1$.

We have then: $2 \cos \alpha + 1 =$ an integer, from which it is seen that the values of $\alpha$ compatible with a lattice base system are: 60$^{\circ}$,90$^{\circ}$, 120$^{\circ}$, 180$^{\circ}$, 240$^{\circ}$, 270$^{\circ}$,300$^{\circ}$, 360$^{\circ}$.

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