= Solution
For torque-free motion, the inertial angular momentum $\mathbf L=I\boldsymbol\omega$ and energy
$T=\tfrac12\boldsymbol\omega\cdot I\boldsymbol\omega$ are constant. Consider the energy ellipsoid
$$
\mathbf x\cdot I\mathbf x=2T,
$$
which is fixed in the body. At its point $\mathbf x=\boldsymbol\omega$, its normal is
$I\boldsymbol\omega=\mathbf L$, and its tangent plane is
$$
\mathbf L\cdot\mathbf x=2T.
$$
This plane has fixed normal and fixed distance $2T/|\mathbf L|$ from the origin, so it is the invariable plane. The material velocity of the contact vector is
$\boldsymbol\omega\times\boldsymbol\omega=0$. Thus the contact point is instantaneously at rest and the ellipsoid rolls without slipping on the plane. This is the <Poinsot construction>.
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