Rigid-body kinetic energy about the centre of mass ($T=\frac{1}{2}M|v_G{|}^2+\frac{1}{2}\omega \cdot L_G$)

Expanding the squared velocity $v_G+\omega \times r$ and integrating over mass removes the mixed term because the centre-of-mass first moment is zero. The remaining rotational integral is half the scalar product of angular velocity and angular momentum about the centre of mass. Principal axes diagonalize that rotational quadratic form.