In Chess, we already have Rooks, Knights, Bishops, and so on. All we need to do is to decide how they move in the third dimension.
This is not always easy, however.
For you, the squares must alternate in color both horizontally and vertically, and a 3D Bishop must be able to move from 1c1 to 2c2 to 3c3 and so on. If you turn the board on its side, the moves look the same.
Unfortunately, the true 3D diagonal is not there in your translation: nothing in the game can move from 1a1 to 2b2 in one turn.
Most forms of 3D Chess based on this translation have cheated and used a damaged translation rather than compromise with symmetry.
Far behind the values of symmetry and proper translation came playability.
Apparently, mathematicians seem to prefer this translation.
If you turn the board on its side, the pieces fall off.
It won't bother you at all if up and down are different from side-to-side; after all, Pawns already only move "forwards", so what's the big deal?
To a chessplayer, sitting behind the board, the 3D move from 1c1 to 2c2 to 3c3 is obviously a Rook move.
My research says that it is possible that nobody else before me realized that the abstract translation was awful, looked further for an answer, and invented the upright translation.
Probably this just shows that my research is bad.