We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$. We now set $\alpha = e _ { 1 } - e _ { 2 }$, $\beta = 2 e _ { 2 }$, $H _ { \alpha } = \left( \begin{array} { c c c c } 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$ and $H _ { \beta } = \left( \begin{array} { c c c c } 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \end{array} \right)$. a) Using the results from question III.C.3, show that there exists a pair $\left( X _ { \alpha } , X _ { - \alpha } \right) \in \mathcal { A } _ { \alpha } \times \mathcal { A } _ { - \alpha }$ and a pair $\left( X _ { \beta } , X _ { - \beta } \right) \in \mathcal { A } _ { \beta } \times \mathcal { A } _ { - \beta }$ such that $( X _ { \alpha } , H _ { \alpha } , X _ { - \alpha } )$ and $( X _ { \beta } , H _ { \beta } , X _ { - \beta } )$ are admissible triples of $\mathcal { A }$. b) Show that $\mathcal { A }$ is the smallest vector subspace of $\mathcal { M } ( 4 , \mathbb { R } )$ stable by bracket and containing the matrices $X _ { \alpha } , H _ { \alpha } , X _ { - \alpha } , X _ { \beta } , H _ { \beta }$ and $X _ { - \beta }$.
We use the notations from Parts I and II as well as from question III.B. We assume
$$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$
and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$.
We now set $\alpha = e _ { 1 } - e _ { 2 }$, $\beta = 2 e _ { 2 }$, $H _ { \alpha } = \left( \begin{array} { c c c c } 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$ and $H _ { \beta } = \left( \begin{array} { c c c c } 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \end{array} \right)$.
a) Using the results from question III.C.3, show that there exists a pair $\left( X _ { \alpha } , X _ { - \alpha } \right) \in \mathcal { A } _ { \alpha } \times \mathcal { A } _ { - \alpha }$ and a pair $\left( X _ { \beta } , X _ { - \beta } \right) \in \mathcal { A } _ { \beta } \times \mathcal { A } _ { - \beta }$ such that $( X _ { \alpha } , H _ { \alpha } , X _ { - \alpha } )$ and $( X _ { \beta } , H _ { \beta } , X _ { - \beta } )$ are admissible triples of $\mathcal { A }$.
b) Show that $\mathcal { A }$ is the smallest vector subspace of $\mathcal { M } ( 4 , \mathbb { R } )$ stable by bracket and containing the matrices $X _ { \alpha } , H _ { \alpha } , X _ { - \alpha } , X _ { \beta } , H _ { \beta }$ and $X _ { - \beta }$.