Solution
= Solution
The <kinetic energy> and squared angular-momentum magnitude are
$$
T=\frac12(I_1\omega_1^2+I_2\omega_2^2+I_3\omega_3^2),
\qquad
L^2=I_1^2\omega_1^2+I_2^2\omega_2^2+I_3^2\omega_3^2.
$$
Differentiate and substitute Euler's equations. In $\dot T$, the coefficient of $\omega_1\omega_2\omega_3$ is
$$
(I_2-I_3)+(I_3-I_1)+(I_1-I_2)=0.
$$
The analogous weighted sum in $dL^2/dt$ also cancels, so both quantities are conserved.