Does such a matrix have to be orthogonal?

Question

$M$ is an $n\times n$ matrix with determinant $1$.

We know that if $M$ preserves euclidean norms for all vectors then $M$ must be orthogonal, that is preserves dot product.

But what if $M$ has the property that there is a basis of vectors $x_i$ with norm $1$ such that each $Mx_i$ also has norm $1$. Does $M$ have to be orthogonal in this case? This looks like it is true in $R^2$.

Answer 1

The answer is no. For instance, consider the matrix $$ M = \pmatrix{1&1\\0&1} $$ and the basis $$ x_1 = (1,0), \quad x_2 = \frac 1{\sqrt 5} (-1 , 2). $$