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Is there any 4D Euler brick?

On May 4th, 2010 tsihonglau says:

Perfect cuboid is related to Euler brick whose edges and face diagonals are all integers. It is know that there are infinite Euler bricks. But is there any 4D Euler brick? In other words, is there any solution to the following system of Diophantine equations:

$ a^2 + b^2 = e^2 $

$ a^2 + c^2 = f^2 $

$ b^2 + c^2 = g^2 $

$ a^2 + d^2 = h^2 $

$ b^2 + d^2 = i^2 $

$ c^2 + d^2 = j^2 $

I computed a, b, c, d up to 1 million with brute force and found no solution. Any idea?

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