Isometric embedding from X to $\mathbb{R}^4$(TIFR GS 2017)

Question

Does there always exist an isometric embedding from $X$ to $\mathbb{R}^4$ for finite $X$.

I think yes, because I think every finite space can be isometrically embedded to a Hilbert space. Any ideas. Thanks.

Answer 1

Hint: Consider a lower dimensional analog. Is this possible for $\Bbb R^2$? What if we have $4$ equidistant points?

Of course, we can embed these points in $\Bbb R^3$ as the vertices of a regular tetrahedron.