At my university, we were to solve this exercise during representation theory lectures.
We consider the space $E$ of traceless Hermitian matrices as a real vector space of dimension 3: $$\begin{bmatrix}x_1&x_2 + ix_3\\x_2 − ix_3&−x_1\end{bmatrix}.$$
Show that the scalar product induced by the ’standard’ norm on $E$ is $P$-invariant, where
$$\begin{align} P: SU_2&\to {\rm Aut}_R(E),\\ P(A) \to X&\mapsto AXA^{−1}. \end{align}$$
My progress:
Take the scalar product on E, which is induced by a standard norm on E. Then check the definition of invariance, which is $\langle f(u), f(v)\rangle_E$ is equal to $\langle u,v\rangle_{SU_2}$ for any $u,v \in E$ and $f: SU_2 \to {\rm Aut}_\mathbb{R}(E) $. My guess is that the scalar product on $E$ the square root of sum of squares of vector coordinates. However, I don´t know how to check the equality of these scalar product, or where I am possibly wrong.
Any advice is welcome.
Usually the scalar product on a set of square complex matrices is introduced by the rule $(x,y)=\operatorname{tr}(xy^*)$, where $y^*=\bar y^T$ (as you can see no square roots).
Here $\operatorname{tr}(x)$ is the trace of matrix $x$, i.e. the sum of elements of the main diagonal of $x$. In the case of Hermitian matrices, the formula is simplified by $(x,y)=\operatorname{tr}(xy)$. So $$ (f(x),f(y))=\operatorname{tr}(f(x)f(y))=\operatorname{tr}(axa^{-1}aya^{-1}) $$
$$ =\operatorname{tr}(axya^{-1})=\operatorname{tr}(xy)=(x,y). $$