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 $$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$ If $L = \left( l _ { i , j } \right) _ { i , j } \in O ( 1 , p ) , v = ( 1,0 , \ldots , 0 )$ and $v ^ { \prime } = ( 0,1,0 , \ldots , 0 )$, give the equations on the $l _ { i , j }$ corresponding to $$\varphi _ { p + 1 } \left( f ( v ) , f \left( v ^ { \prime } \right) \right) = \varphi _ { p + 1 } \left( v , v ^ { \prime } \right) , \quad q _ { p + 1 } ( f ( v ) ) = q _ { p + 1 } ( v ) \quad \text { and } \quad q _ { p + 1 } \left( f \left( v ^ { \prime } \right) \right) = q _ { p + 1 } \left( v ^ { \prime } \right)$$ What do we obtain similarly with ${ } ^ { t } L$ ?
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
$$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$
If $L = \left( l _ { i , j } \right) _ { i , j } \in O ( 1 , p ) , v = ( 1,0 , \ldots , 0 )$ and $v ^ { \prime } = ( 0,1,0 , \ldots , 0 )$, give the equations on the $l _ { i , j }$ corresponding to
$$\varphi _ { p + 1 } \left( f ( v ) , f \left( v ^ { \prime } \right) \right) = \varphi _ { p + 1 } \left( v , v ^ { \prime } \right) , \quad q _ { p + 1 } ( f ( v ) ) = q _ { p + 1 } ( v ) \quad \text { and } \quad q _ { p + 1 } \left( f \left( v ^ { \prime } \right) \right) = q _ { p + 1 } \left( v ^ { \prime } \right)$$
What do we obtain similarly with ${ } ^ { t } L$ ?