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 assume that the conditions of questions 4 and 5 are satisfied and that $\lambda(0) = 0, \mu(0) = 1$. We denote by $\mathbf{C}[X]$ the algebra of polynomials with complex coefficients in one indeterminate $X$.
8a. Show that $\mathbf{C}[E]$ is isomorphic (as an algebra) to $\mathbf{C}[X]$.
8b. Show that $\mathbf{C}[F]$ is isomorphic (as an algebra) to $\mathbf{C}[X]$.
8c. Show that $\mathbf{C}[H]$ is isomorphic (as an algebra) to $\mathbf{C}[X]$.

In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ If $A$ and $B$ are two matrices in $\mathcal{M}_n(\mathbf{K})$, their Lie bracket is defined by $[A, B] = AB - BA$. In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{13}$ ▷ Deduce from question 12) that $\mathcal{A}_G$ is stable under the Lie bracket, i.e. $$\forall A \in \mathcal{A}_G, \forall B \in \mathcal{A}_G, [A, B] \in \mathcal{A}_G.$$

Show that the set of shift-invariant endomorphisms of $\mathbb{K}[X]$ is a subalgebra of $\mathcal{L}(\mathbb{K}[X])$. Is the set of delta endomorphisms of $\mathbb{K}[X]$ closed under addition? under composition?

a) Show that $\mathbb{H}$ is a sub-$\mathbb{R}$-algebra of $M_2(\mathbb{C})$ stable under $Z \mapsto Z^*$. b) Let $Z \in \mathbb{H}$. Calculate $ZZ^*$ and deduce that every non-zero element of $\mathbb{H}$ is invertible. c) Let $Z \in \mathbb{H}$. Show that $Z \in \mathbb{R}_{\mathbb{H}}$ if and only if $ZZ' = Z'Z$ for all $Z' \in \mathbb{H}$.

Show that $\mathrm{Aut}(\mathbb{H})$ is a subgroup of $\mathrm{GL}(\mathbb{H})$, containing $\alpha(u,u)$ for all $u \in S$.

We fix an element $i_A$ of $A$ such that $i_A^2 = -1$. We denote $U = \mathbb{R} + \mathbb{R}i_A$ and we define the map $$T : A \rightarrow A,\quad T(x) = i_A x i_A.$$ We denote $\mathrm{id} : A \rightarrow A$ the identity map of $A$. a) Show that $T(xy) = -T(x)T(y)$ for all $x, y \in A$. b) Calculate $T^2 = T \circ T$ and deduce that $A = \ker(T - \mathrm{id}) \oplus \ker(T + \mathrm{id})$.

Show that $\ker(T + \mathrm{id}) = U$ and deduce that $\ker(T - \mathrm{id}) \neq \{0\}$.

We fix $\beta \in \ker(T - \mathrm{id}) \setminus \{0\}$. a) Show that the map $x \mapsto \beta x$ sends $\ker(T - \mathrm{id})$ into $\ker(T + \mathrm{id})$. Deduce that $\beta^2 \in U$ and that $\ker(T - \mathrm{id}) = \beta U$. b) Show that $\beta^2 \in ]-\infty, 0[$. c) Prove Theorem B: An algebraic $\mathbb{R}$-algebra without zero divisors is isomorphic to $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$.

Let $A$ be a $\mathbb{R}$-algebra such that there exists a norm $\|\cdot\|$ on the $\mathbb{R}$-vector space $A$ satisfying $$\forall x, y \in A,\quad \|xy\| = \|x\| \cdot \|y\|.$$ Let $x, y \in A$ such that $xy = yx$ and such that $V = \mathbb{R}x + \mathbb{R}y$ is of dimension 2 over $\mathbb{R}$. Show that $$\forall u, v \in V \quad \|u+v\|^2 + \|u-v\|^2 \geq 4\|u\| \cdot \|v\|$$ and that the restriction of $\|\cdot\|$ to $V$ comes from an inner product on $V$.

Let $A$ be a $\mathbb{R}$-algebra such that there exists a norm $\|\cdot\|$ on the $\mathbb{R}$-vector space $A$ satisfying $$\forall x, y \in A,\quad \|xy\| = \|x\| \cdot \|y\|.$$ Show that $x^2 \in \mathbb{R} + \mathbb{R}x$ for all $x \in A$. One may use the result from question 25 with $y = 1$.