(\mathbf{R}-e^{i\phi}\mathbf{I})\mathbf{u}_1 = 0 implies that also (\mathbf{R}-e^{-i\phi}\mathbf{I})\mathbf{u}^*_1 = 0 holds, so \mathbf{u}_2 = \mathbf{u}^*_1 can be chosen for \mathbf{u}_2 . Similarly, (\mathbf{R}-\mathbf{I})\mathbf{u}_3 = 0 can result in a \mathbf{u}_3 with real entries only, for a proper rotation matrix \mathbf{R}.
