\in D $ into $$g ( {\theta}, u , v ) :=  \bigl( 1 - u^2 - v^2 \bigr) ^{-1/2}
\left( \begin{matrix} \cos ( {\theta}) + u   & - \sin ( {\theta}) + v \\ \sin (
    {\theta}) + v & \cos ( {\theta}) - u  \end{matrix} \right)$$ where $D = \{(u,v) / \;   u^2 + v^2
< 1 \}$.

Let $E$ be the positive polarization generated by $e - i f$ and let ${\operatorname{Mp}}(S,E)$ be the associated metaplectic group. Let us parametrize ${\operatorname{Mp}}(S,E)$ by $S^1 \times D$ $$( {\theta}, u,v) \rightarrow  ( g ( - 2 {\theta}, u, v) , e^{i {\theta}} ( 1
- u ^2 - v ^2 )^{-\frac{1}{4}} )$$ Then ${\operatorname{Mp}}_* (S,E)$ is the image of $\bigl\{ \cos ( 2 {\theta}) \neq \bigl( 1 -
u ^2 - v^2 \bigr)^{1/2} \bigr\}$. We find easily the index of any element of $ {\operatorname{Sp}}_* (S)$ by computing explicitly the index of one element in each component. cf. figure \[fig:index\].

Observe that when the argument of $z$ runs over an interval $\bigl[ k
\tfrac{\pi}{2}, (k+1) \tfrac{\pi}{2} \bigr[$ with $k \in {{\mathbb{Z}}}$, the index $m(g,z)$ takes two distinct values depending on the sign of $\operatorname{tr}(g) -
2$. Checking the various 