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Let $\mathbb R$ be the field of real numbers and $\mathbb C$ the field of complex numbers. It is well known that that $\mathbb C$ can be embedded in $M_2(\mathbb R)$by the matrix- valued funtion

\begin{equation}M(a+ib)=\left[\begin{array}{cc} a & b \ -b & a \ \end{array}\right].\end{equation}.

This embedding can be extended in the obvious way to an embedding function $\varphi : M_n(\mathbb C)\rightarrow M_{2n}(\mathbb R)$.

My question is: consider the embedding $\varphi :M_2(\mathbb C)\rightarrow M_{4}(\mathbb R)$, it is possible to find an inner automorphism $\Psi : M_4(\mathbb R)\rightarrow M_{4}(\mathbb R)$ such that the intersection $$\varphi (M_2(\mathbb C))\cup C))\cap \Psi(\varphi (M_2(\mathbb C)))= { \alpha I\;\;: \alpha\in\mathbb R}?$$

where $I$ C)))$$

is the identity matrix.scalar matrices in $M_4(\mathbb R)$?
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Let $\mathbb R$ be the field of real numbers and $\mathbb C$ the field of complex numbers. It is well known that that $\mathbb C$ can be embedded in $M_2(\mathbb R)$ by the matrix- valued funtion

\begin{equation}M(a+ib)=\left[\begin{array}{ccc} begin{equation}M(a+ib)=\left[\begin{array}{cc} a & b \ -b & a \ \end{array}\right].\end{equation}

This embedding can be extended in the obvious way to an embedding function $\varphi M_n(\mathbb C)\rightarrow M_{2n}(\mathbb R)$.

My question is: consider the embedding $\varphi :M_2(\mathbb C)\rightarrow M_{4}(\mathbb R)$, it is possible to find an inner automorphism $\Psi : M_4(\mathbb R)\rightarrow M_{4}(\mathbb R)$ such that

$$\varphi (M_2(\mathbb C))\cup \Psi(\varphi (M_2(\mathbb C)))= { \alpha I\;\;: \alpha\in\mathbb R}?$$

where $I$ is the identity matrix.
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Inner Automorphisms of Matrix Algebras

Let $\mathbb R$ be the field of real numbers and $\mathbb C$ the field of complex numbers. It is well known that that $\mathbb C$ can be embedded in $M_2(\mathbb R)$ by the matrix- valued funtion

\begin{equation}M(a+ib)=\left[\begin{array}{ccc} a & b \ -b & a \end{array}\right].\end{equation}

This embedding can be extended in the obvious way to an embedding function $\varphi M_n(\mathbb C)\rightarrow M_{2n}(\mathbb R)$.

My question is: consider the embedding $\varphi :M_2(\mathbb C)\rightarrow M_{4}(\mathbb R)$, it is possible to find an inner automorphism $\Psi : M_4(\mathbb R)\rightarrow M_{4}(\mathbb R)$ such that

$$\varphi (M_2(\mathbb C))\cup \Psi(\varphi (M_2(\mathbb C)))= { \alpha I\;\;: \alpha\in\mathbb R}?$$

where $I$ is the identity matrix.