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 $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$. Show that $\mathcal { A }$ is a vector subspace of $\mathcal { M } ( 4 , \mathbb { R } )$ stable by bracket. Show that we have $\mathcal { A } _ { 0 } = \mathcal { E }$, where $\mathcal { A } _ { 0 }$ denotes $\mathcal { A } _ { \lambda }$ when $\lambda$ is the zero linear form. Give a basis of $\mathcal { A } _ { 0 }$.
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 $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$.
Show that $\mathcal { A }$ is a vector subspace of $\mathcal { M } ( 4 , \mathbb { R } )$ stable by bracket. Show that we have $\mathcal { A } _ { 0 } = \mathcal { E }$, where $\mathcal { A } _ { 0 }$ denotes $\mathcal { A } _ { \lambda }$ when $\lambda$ is the zero linear form. Give a basis of $\mathcal { A } _ { 0 }$.