Proof: Roots $I^{(a)}$ are Real

$$I_{ik}\omega_k=I\omega_i$$     for omega along principle axis 

$$I_{ik}\omega_k\omega_i^*=I\omega_i\omega_i^* \omega_i^*$$

$$I_{ki}\omega_i=I\omega_k$$    same equation with k

$$I_{ki}^*\omega_i^*=I^*\omega_k^*$$     take CC

$$I_{ki}^*\omega_i^*\omega_k=I^*\omega_k^*\omega_k \omega_k$$

$$I_{ki}\omega_k\omega_i^*=I^*\omega_k^*\omega_k$$     inertia tensor is real, omegas commute 

$$I_{ik}\omega_k\omega_i^*=I^*\omega_i^*\omega_i$$     I symmetric, replace dummy rhs

$$(I-I^*)\omega_i\omega_i^*=0$$    lhs same as 2nd line

$$(I-I^*)=0$$

So the roots must be real.