We assume in this part that $\mathbb { K } = \mathbb { R }$. Let $q \in \mathcal { Q } \left( \mathbb { R } ^ { n } \right) ( n \geq 1 )$. Prove that there exists a pair of integers $( r , s ) ( r + s = n )$ such that $q$ is isometric to $Q _ { r , s }$ defined on the canonical basis of $\mathbb { R } ^ { n }$ by
$$Q _ { r , s } \left( x _ { 1 } , \ldots , x _ { n } \right) = \sum _ { i = 1 } ^ { r } x _ { i } ^ { 2 } - \sum _ { j = r + 1 } ^ { n } x _ { j } ^ { 2 }$$