Euler bricks and the $4^{th}$ dimension

Question

I've heard of Euler bricks, which go like this:

A Euler brick is a brick that has 3 sides, and any combination of the sides using the Pythagorean theorem will get a whole number. the Pythagorean theorem is:

$$ A^2+B^2=C^2 \text{ or } \sqrt{A^2+B^2}=C$$

I am wondering... is there any 4D Euler brick? as in A, B, C, D, E, F, G, H, I, J, where

$$ \begin{align} a^2+b^2&=e^2\\ a^2+c^2&=f^2\\ a^2+d^2&=g^2\\ b^2+c^2&=h^2\\ b^2+d^2&=i^2\\ c^2+d^2&=j^2\\ \end{align} $$

Thanks in advance.

Answer 1

The existence of $4$-dimensional Euler bricks is still an open problem. For the same question see here.