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 $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$. We assume that the vector $a$ is non-zero.
Deduce from question III.E.1 that there exists an element $L _ { 1 }$ of $G$ such that: $$L _ { 1 } L = \left( \begin{array} { c c c c } \ell _ { 1,1 } & \ell _ { 1,2 } & \ell _ { 1,3 } & \ell _ { 1,4 } \\ \alpha & \lambda _ { 1 } & \lambda _ { 2 } & \lambda _ { 3 } \\ 0 & \mu _ { 1 } & \mu _ { 2 } & \mu _ { 3 } \\ 0 & \nu _ { 1 } & \nu _ { 2 } & \nu _ { 3 } \end{array} \right)$$ where $\alpha$ is a strictly positive real number that we will specify, $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } , \mu _ { 1 } , \mu _ { 2 } , \mu _ { 3 } , \nu _ { 1 } , \nu _ { 2 }$ and $\nu _ { 3 }$ are real numbers that we will not seek to determine.