Suppose $\{ v_1, \ldots, v_k \}$ is a set of vectors in $\mathbb{R}^n$. The associated $k\times k$ Gram matrix is given by $$ G = [v_i \cdot v_j]_{i,j} $$ It's (apparently) well known that the Gram matrix of a set of vectors determines the vectors up to an isometry of $\mathbb{R}^n$ (e.g. [1])
My question is: does anyone know of a reference for an algorithm that performs the reconstruction? More precisely, I'm looking for an algorithm that takes $G$ as input and outputs $\{ Av_1, \ldots, Av_k \}$ for some $A \in SO(n)$.