Equilateral triangles in Z⁴

We give a characterization of all three points in $\mathbb R^4$ with integer coordinates which are at the same Euclidean distance apart. In three dimension the problem is characterized in terms of solutions of the Diophantine equations $a^2+b^2+c^2=3d^2$. In $\mathbb R^4$, the characterization is essentially based on two different solutions of the same equation.