A right* square based pyramid has vertical height 4 units, and the diagonal of its square base is 6 units long.  *This means that the line from the vertex to the middle of the square base is perpendicular to the base,

A cuboid is formed by making a cut parallel to the base of the pyramid so that the top of the resulting solid is a square. Four more cuts are then made at right angles to the top and bottom of this new solid, so that the square top is one face of a cuboid that could just fit inside the original pyramid.

What is the largest possible volume of a cuboid formed in this way?