I'm interested in seeing if a symmetric matrix $S$ (which represents scalings/skews in the standard Euclidean scalar product), viewed through the lens of a non Euclidean scalar product (with positive definite Gram matrix $M$), can ever be a special orthogonal matrix $R$ (i.e. a pure rotation in the Euclidean scalar product).
So can $$MS = R$$ ($M$ positive definite, $S$ symmetric, $R$ special orthogonal) ever be true, if not, is there a simple proof of why not? If so, is there a nice example?