Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 }$ and $L ^ { \prime } = \left( \ell _ { i , j } ^ { \prime } \right) _ { 1 \leqslant i , j \leqslant 4 }$ be two elements of $\tilde { O } ( 1,3 )$. We set $L ^ { \prime \prime } = L L ^ { \prime } = \left( \ell _ { i , j } ^ { \prime \prime } \right) _ { 1 \leqslant i , j \leqslant 4 }$.
Prove the following inequalities:
$$0 \leqslant \sqrt { \sum _ { k = 2 } ^ { 4 } \ell _ { 1 , k } ^ { 2 } } \sqrt { \sum _ { k = 2 } ^ { 4 } \ell _ { k , 1 } ^ { \prime 2 } } + \sum _ { k = 2 } ^ { 4 } \ell _ { 1 , k } \ell _ { k , 1 } ^ { \prime } < \ell _ { 1,1 } ^ { \prime \prime }$$
Deduce that the set $\tilde { O } ( 1,3 )$ is a subgroup of the Lorentz group $O ( 1,3 )$.