We assume $\mathbb{K} = \mathbb{R}$. For $f \in O \left( Q _ { 2,1 } \right)$, we denote by $( x _ { f } , y _ { f } , z _ { f } )$ the vector $f ( 0,0,1 )$. We also denote by $S O _ { 2,1 } ^ { + } : = \left\{ M = j ( f ) \in S O _ { 2,1 } \mid z _ { f } > 0 \right\}$.
(a) Prove that, for all $t \in \mathbb { R }$, the linear map $r _ { t }$ whose matrix (in the canonical basis of $\mathbb { R } ^ { 3 }$ ) equals $$\left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \operatorname { ch } ( t ) & \operatorname { sh } ( t ) \\ 0 & \operatorname { sh } ( t ) & \operatorname { ch } ( t ) \end{array} \right]$$ is in $S O _ { 2,1 } ^ { + }$ (one may subsequently call such a linear map a hyperbolic rotation).
(b) Let $M = j ( f )$. Suppose that $M \in S O _ { 2,1 } ^ { + }$. Show that there exists a rotation (in the usual sense) $\rho$ with axis $( 0,0,1 )$ and $t \in \mathbb { R }$ such that $r _ { t } \circ \rho \circ f \in S O _ { 2,1 } ^ { + }$ and satisfies $r _ { t } \circ \rho \circ f ( 0,0,1 ) = ( 0,0,1 )$.
(c) Prove that $S O _ { 2,1 } ^ { + }$ is path-connected.