Two reflections at axes/Composition not diagonalizable/Example

Let ${}G_{1}$ and ${}G_{2}$ denote two lines in ${}\mathbb {R} ^{2}$ through the origin, and let ${}\varphi _{1}$ and ${}\varphi _{2}$ 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

${}\psi =\varphi _{2}\circ \varphi _{1}\,$

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.