10x1 in a cube - G4G10 puzzle

For the Tenth Gathering For Gardner (login credentials required) I designed an appropriate exchange gift. It consists of 10 identical pieces, shaped like a figure 1. These pieces pack into a 4x4x4 cube in 4 ways.
drawing of the shape of each piece
An nice additional property is that this shape tiles the plane. This minimizes the amount of waste generated while laser-cutting the pieces.

The puzzle is not easy. Please try to solve it manually before resorting to the solutions.

Solutions of the puzzle

There are four solutions, depicted below. Each row of four grids depicts the four layers of one solution
Solution 1

Solution 2

Solution 3

Solution 4
The puzzle was designed with BurrTools.

Peter Knoppers