Past exam of the physics B course of the University of Cambridge / 2014 / Paper 2 / Section B / B8 / b

For a rigid body about a chosen origin,The inertia tensor is real and symmetric, so orthogonal principal axes diagonalize it. Angular momentum is parallel to angular velocity for rotation about a principal axis, but need not be parallel in a general direction. Its eigenvalues are the principal moments of inertia.

If the origin is translated from the centre of mass by $\mathbf{a}$, the parallel-axis theorem becomes $I_{ij}=I_{ij}^{\mathrm{CM}}+M(a^2{\delta }_{ij}-a_i{a}_j)$. In body axes, Euler's equation is $\mathbf{\tau }=(d\mathbf{L}/dt{)}_{\mathrm{body}}+\mathbf{\omega }\times \mathbf{L}$. Thus the full tensor is needed whenever the rotation is not confined to one fixed principal axis.