= Solution
Multiply the three <Euler equations for a torque-free rigid body> by $\omega_1,\omega_2,\omega_3$ and add. The cubic terms cancel, giving conservation of the <rotational kinetic energy>
$$
\boxed{E=\frac12\left(I_1\omega_1^2+I_2\omega_2^2+I_3\omega_3^2\right).}
$$
Instead multiply the equations by $I_1\omega_1,I_2\omega_2,I_3\omega_3$ and add. The cubic terms again cancel, giving
$$
\boxed{L^2=I_1^2\omega_1^2+I_2^2\omega_2^2+I_3^2\omega_3^2,}
$$
the squared magnitude of the conserved <angular momentum>. For an asymmetric body these two quadratic forms are not proportional, so the <torque-free rigid-body invariants> are independent.
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