= Solution
Reflection symmetry of the cube in each coordinate plane makes every off-diagonal integral
$$
-\rho\int_Bx_ix_j\,dV\qquad(i\ne j)
$$
vanish because its integrand is <odd function>[odd] in one coordinate. Permuting the three coordinate axes leaves the cube unchanged, so its three diagonal moments are equal. Thus its <inertia tensor> at the centre has the isotropic form
$$
I_{ij}(\mathbf0)=\lambda\delta_{ij}.
$$
For
$$
\mathbf a=\frac\ell2(1,1,0),
\qquad
|\mathbf a|^2=\frac{\ell^2}{2},
$$
part (a) and $\lambda=M\ell^2/6$ give
$$
\boxed{
I(\mathbf a)
=\frac{M\ell^2}{12}
\begin{pmatrix}
5&-3&0\\
-3&5&0\\
0&0&8
\end{pmatrix}}.
$$
The displayed symmetric matrix has orthogonal <eigenvectors>
$$
(1,1,0),\qquad(1,-1,0),\qquad(0,0,1)
$$
with corresponding <eigenvalues>
$$
\frac{M\ell^2}{6},
\qquad
\frac{2M\ell^2}{3},
\qquad
\frac{2M\ell^2}{3}.
$$
For a unit angular velocity, $|\mathbf L|=|I\boldsymbol\omega|$ is minimized and maximized along eigenvectors for the smallest and largest eigenvalues. Their ratio is
$$
\boxed{
\frac{|\mathbf L|_{\max}}{|\mathbf L|_{\min}}
=\frac{2M\ell^2/3}{M\ell^2/6}=4}.
$$
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