 Off-axis angular momentum of a square lamina

For a uniform square of side $a$, the principal inertia tensor is $\mathrm{diag}(m{a}^2/12,m{a}^2/12,m{a}^2/6)$. A rotation axis tilted $45^{\circ }$ to the surface has equal in-plane and normal angular-velocity components. Multiplying each by its own principal moment gives the stated magnitude. In general $I_{\mathrm{axis}}\omega$ is only the axial projection of angular momentum.