Preliminaries
Let $\mathbb E ^ n$ denote the elliptical space of order $n$: $$ \mathbb E ^ n = \mathbb S ^ n / \sim$$ where $\sim$ is an equivalence relation identifying $x$ and $-x$. This can also be thought of as the real projective space of order n $\mathbb{RP}^n$ by identifying lines through the origin with their intersection with $\mathbb S ^n$. Equip $\mathbb E^n$ with the following metric $\varepsilon$: $$ \varepsilon([x], [y]) = \arccos(\lvert \langle x, y \rangle \rvert) .$$ I want to show that the isometries of this space are all induced by isometries of the n-sphere $\mathbb S ^ n$ (e.g. by the orthogonal group $O(n+1)$). For example, let $U \in O(n+1)$ be an orthogonal matrix. Then $U$ induces an isometry $f_U$ of $\mathbb E ^ n$ like this: $$ f_U([x]) = [Ux].$$ Clearly this is an isometry of $\mathbb E ^n$, as orthogonal matrices preserve the inner product.
What have I done
Let $f$ be an isometry of $\mathbb E ^n$ and denote by $e_i$ the i-th unit vector. Choose $w_i \in f([e_i])$. Clearly $\{w_1, \dots, w_{n+1} \}$ is an orthonormal base of $\mathbb R ^ {n+1}$. Here is where I am stuck. Obviously it matters which $w_i$ we choose, as for each $i$ we have two choices. We want to choose the $w_i$ such that for the matrix $W=\left( w_1, \dots , w_{n+1}\right)$ we have $f=f_W$. But I can't really make out a criteria on how to choose the $w_i$.
After the $w_i$ are choosen correctly we also have to show that $f([x]) = [Wx]$ for all $x\in \mathbb S ^ n$. This is easy to show for the unit vectors, but I have no idea how to show it in general, as there is no notion of linearity for $f$. One can probably use the isometric property of $f$ for $x = \sum x_i e_i$ and $y = \sum y_i w_i \in f([x])$ $$\varepsilon (f([x]), f([e_i])) = \varepsilon([x], [e_i])$$ canceling the $\arccos$ gives us $$\lvert y_i \rvert = \lvert \langle y, w_i \rangle \rvert = \lvert \langle x, e_i \rangle \rvert = \lvert x_i \rvert.$$