If by taking a thin rod, and finding its Moment of Inertia about an axis, say through the mid point of its side, one can observe that stretching the rod uniformly along the axis of rotation will give the shape of a uniform rectangle whose MOI is the same with the thin uniform rod. What happens when you decide to go back to these first principles in knowing the MOI of a rectangle about its diagonal?
Let the red axis be the $s$ axis and draw another axis through the centre parallel to the plane of the lamina and perpendicular to the $s$ axis. Call this the $t$ axis.
By symmetry, $I_s=I_t$.
By Perpendicular Axes, $$I_s+I_t=I_z,$$ where $z$ is the axis perpendicular to the plane of the lamina and through the centre.
You already have the result that $I_z=\frac 23ma^2$ by similar consideration of the MI about a diameter and the Perpendicular Axes Theorem.
Therefore the MI about a diagonal is $$\frac 13ma^2$$