We denote, for every real $\gamma$, $L ( \gamma ) = \left( \begin{array} { c c } \operatorname { ch } \gamma & \operatorname { sh } \gamma \\ \operatorname { sh } \gamma & \operatorname { ch } \gamma \end{array} \right)$.
Show, for all real numbers $\gamma$ and $\gamma ^ { \prime }$, the equality:
$$L ( \gamma ) L \left( \gamma ^ { \prime } \right) = L \left( \gamma + \gamma ^ { \prime } \right)$$
Deduce that $O ^ { + } ( 1,1 ) \cap \tilde { O } ( 1,1 )$ is a commutative subgroup of the group $O ^ { + } ( 1,1 )$.