Two reflections at axes/Composition not diagonalizable/Example

Let $G_1$ and $G_2$ denote two lines in ${ℝ}^2$ through the origin, and let ${\phi }_1$ and ${\phi }_2$ denote the reflections at these axes. A reflection at an axis is always diagonalizable, the axis and the line orthogonal to the axis are eigenlines (with eigenvalues $1$ and $-1$). The composition

$\psi ={\phi }_2\circ {\phi }_1$

of the reflections is a plane rotation, the angle of rotation being twice the angle between the two lines. However, a rotation is only diagonalizable if the angle of rotation is $0$ or $180$ degree. If the angle between the axes is different from $0,90$ degree, then $\psi$ does not have any eigenvector.