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I got stuck on some fact concerning the Cartan immersion of symmetric spaces:

Let $M$ be a Riemann symmetric space associated to the Lie group $G$ with involutive automorphism $\sigma$. Let $G$ act on itself by $g \cdot h= gh \sigma(g^{-1})$. Then one can consider the orbit $P:=\{g \sigma(g^{-1})\}_{g \in G}$ of the neutral element $e \in G$. It is well-known that we can get a map $f: M \rightarrow P$, which is called the Cartan immersion of $M$. Up to here I can follow, but now I have a question concerning the behaviour of certain metrics:

If the given metric on the tangent space $T_e G$ is invariant under the action of all elements in the fixed point set of $\sigma$, then one construct a metric by transporting it via the action of $G$ on itself. One can also construct a metric on $M$ by transporting via the left multiplication by $G$. Now I got stuck on the following: It is said in my lecture notes that under the map $f$ we have that $\|df(v)\|=2\|v\|$ for elements $v$ in some tangent pace of $M$, i.e. the norm is doubled under $f$, when considering the two metrics defined above. I do not see why this holds, if we define the metrics as above and in my notes it is said that this can be seen trivially. Can you maybe tell me, why it is true?

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