We define $$\varphi _ { p + 1 } \left( v , v ^ { \prime } \right) = { } ^ { t } V \Delta _ { p + 1 } V ^ { \prime } = v _ { 1 } v _ { 1 } ^ { \prime } - \sum _ { i = 2 } ^ { p + 1 } v _ { i } v _ { i } ^ { \prime }$$ and $$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$ Let $L \in \mathcal { M } _ { p + 1 } ( \mathbb { R } )$ and $f$ the endomorphism of $\mathbb { R } ^ { p + 1 }$ canonically associated. Show that the following three assertions are equivalent: i. $L \in O ( 1 , p )$; ii. $\forall \left( v , v ^ { \prime } \right) \in \left( \mathbb { R } ^ { p + 1 } \right) ^ { 2 } , \varphi _ { p + 1 } \left( f ( v ) , f \left( v ^ { \prime } \right) \right) = \varphi _ { p + 1 } \left( v , v ^ { \prime } \right)$; iii. $\forall v \in \mathbb { R } ^ { p + 1 } , q _ { p + 1 } ( f ( v ) ) = q _ { p + 1 } ( v )$.
We define
$$\varphi _ { p + 1 } \left( v , v ^ { \prime } \right) = { } ^ { t } V \Delta _ { p + 1 } V ^ { \prime } = v _ { 1 } v _ { 1 } ^ { \prime } - \sum _ { i = 2 } ^ { p + 1 } v _ { i } v _ { i } ^ { \prime }$$
and
$$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$
Let $L \in \mathcal { M } _ { p + 1 } ( \mathbb { R } )$ and $f$ the endomorphism of $\mathbb { R } ^ { p + 1 }$ canonically associated.
Show that the following three assertions are equivalent:
i. $L \in O ( 1 , p )$;
ii. $\forall \left( v , v ^ { \prime } \right) \in \left( \mathbb { R } ^ { p + 1 } \right) ^ { 2 } , \varphi _ { p + 1 } \left( f ( v ) , f \left( v ^ { \prime } \right) \right) = \varphi _ { p + 1 } \left( v , v ^ { \prime } \right)$;
iii. $\forall v \in \mathbb { R } ^ { p + 1 } , q _ { p + 1 } ( f ( v ) ) = q _ { p + 1 } ( v )$.