We set
$$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$
Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ and $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$.
Show that, if the vector $a$ is zero, then the matrix $L$ belongs to the group $G$.