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 $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$, with $\mathcal{R} = \left\{ e _ { 1 } - e _ { 2 } , e _ { 2 } - e _ { 1 } , e _ { 1 } + e _ { 2 } , - e _ { 1 } - e _ { 2 } , 2 e _ { 1 } , - 2 e _ { 1 } , 2 e _ { 2 } , - 2 e _ { 2 } \right\}$. Establish the relation $\mathcal { A } = \mathcal { A } _ { 0 } \oplus \bigoplus _ { \alpha \in \mathcal { R } } \mathcal { A } _ { \alpha }$.
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 $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$, with $\mathcal{R} = \left\{ e _ { 1 } - e _ { 2 } , e _ { 2 } - e _ { 1 } , e _ { 1 } + e _ { 2 } , - e _ { 1 } - e _ { 2 } , 2 e _ { 1 } , - 2 e _ { 1 } , 2 e _ { 2 } , - 2 e _ { 2 } \right\}$.
Establish the relation $\mathcal { A } = \mathcal { A } _ { 0 } \oplus \bigoplus _ { \alpha \in \mathcal { R } } \mathcal { A } _ { \alpha }$.