From what I understand, an orthogonal matrix is one that satisfies $A\cdot A^t = I_n$. In such case, from what I saw online, $\forall x\in \mathbb{R}^n,\;\left\Vert Ax \right\Vert =\left\Vert x \right\Vert $.
I took this to the test and looked at the orthogonal matrix $$V = \left(\begin{smallmatrix}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & -\frac{1}{\sqrt{3}}\\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}}\\ 0 & \frac{2}{\sqrt{6}} & \frac{1}{\sqrt{3}} \end{smallmatrix}\right)$$
(and you may check that this indeed is an orthogonal matrix that satisfies $V\cdot V^t = I_n$)
and looked at $$V\cdot \left(\begin{matrix}1\\ \:1\\ \:1\end{matrix}\right) =\begin{pmatrix}\frac{3\sqrt{2}+\sqrt{6}-2\sqrt{3}}{6}\\ \frac{3\sqrt{2}-\sqrt{6}+2\sqrt{3}}{6}\\ \frac{1+\sqrt{2}}{\sqrt{3}}\end{pmatrix}$$
and obviously with respect to the norm $\left\Vert \cdot \right\Vert _\infty = \max\{x_1,x_2,x_3\}$ we get that $\left\Vert Vx \right\Vert \neq\left\Vert x \right\Vert $.
I assume, therefore, that this property only applies to the standard inner product (dot product).
My question is, is there a definition $\star$ for orthogonal matrices with respect to different inner products (or different norms)? will there be other equivalent definitions similar to $A\cdot A^t = I_n$?
$\star$ I know I can define such matrices by myself, but I'd like to know if such theory has been developed already and what can be said about it, mainly with respect to properties similar to $A\cdot A^t = I_n$.
Also, I do know that for general linear transformation there are orthogonal operators with respect to general inner products, but when converting those to matrices, the lengths seem to be preserved with respect to the inner product.