We assume $\mathbb{K} = \mathbb{R}$. Let $H : = \left\{ ( x , y , z ) \in \mathbb { R } ^ { 3 } \mid z ^ { 2 } = x ^ { 2 } + y ^ { 2 } + 1 \right\}$ be a hyperboloid of two sheets.
(a) Prove that if $f \in O \left( Q _ { 2,1 } \right)$, then $f ( H ) = H$.
(b) We denote by $S O _ { 2,1 } : = \left\{ M \in O _ { 2,1 } \mid \operatorname { det } ( M ) = 1 \right\}$. Prove that $S O _ { 2,1 }$ is a closed subgroup of $O _ { 2,1 }$.