= Solution
For the <inertia tensor of a uniform cube>, reflection <symmetry> makes every off-diagonal element of the central <inertia tensor> vanish. Permutation of the three axes makes its diagonal elements equal. Since $M=8\rho a^3$ and the mean of each squared coordinate in the <cube> is $a^2/3$,
$$
I_{xx}=\int(y^2+z^2)\,dm=\frac23Ma^2,\qquad
\mathsf I=\frac23Ma^2\mathsf1.
$$
Every central axis is therefore a <principal axis>, including the body diagonal. The <angular momentum> is parallel to the <angular velocity>, and its magnitude is
$$
\boxed{J=\frac{16}{3}\rho a^5\omega.}
$$
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