Orthogonal transformation in a nonorthonormal basis ($A^{T}GA=G$)

Let $b_1,\dots ,b_n$ be an arbitrary basis of a real inner product space, with Gram matrix $G=(⟨b_i,b_j⟩)$. If a linear map $T$ has matrix $A$ in this basis, thenConsequently $T$ is an orthogonal transformation exactly when $A^{T}GA=G$. The simpler orthogonal matrix condition $A^{T}A=I$ applies directly when the chosen basis is an orthonormal basis, so that $G=I$.

For example, take a two-dimensional vector space with weighted Euclidean inner product having $G=\mathrm{diag}(1,4)$. Thensatisfies $A^{T}GA=G$, but $A^{T}A=\mathrm{diag}(1/4,4)\ne I$. Thus it represents an orthogonal transformation even though it is not an orthogonal matrix.