The following are the 8 tetracubes which can be made by combining 4 cubes. In all, 32 unit cubes are used. That would form two layers of a 4x4x4 cube.

If you are able to use each tetracube once to form this arrangement I'd be pleased to receive your solution. If it can't be done indicate why.

Note: in the graphic F4, G4 and H4 are above the cubes that form the lower layer. Also, the G and H tetracubes 'screw' one to the left and the other to the right.

A solution might take the form of 2 4x4 grids using letters only; numerals really only assist in orienting oneself.