Show that an embedding sends a conjugate transpose to transpose.

Question

Here is the question I am trying to tackle:

Prove that $$\mathbb C \to M_2(\mathbb R),$$ defined by $$x + iy \mapsto \begin{pmatrix} x & -y \\ y & x\end{pmatrix}$$ defines an embedding. Show also that this embedding sends "conjugate transpose" to "transpose" and multiplication to multiplication.

My definition for embedding is:

My definition of a submanifold is:

My definition of a diffeomorphism is:

A hint was given in the book to consider the chart $x: M_2(\mathbb R) \to \mathbb R^4$ given by $$x\Big(\begin{pmatrix} a & b \\ c & d\end{pmatrix} \Big) = (a,b, a -d, b+c)$$ but I do not know how to use this hint.

Also still I do not how to prove that this is a submanifold, could anyone help me in proving this please?

EDIT:

Note that I am asking here about the proof of a submanifold not about the inverse of the function in the other suggested question.

Answer 1

If$$x\left(\begin{bmatrix}a&b\\c&d\end{bmatrix}\right)=(a,b,a-d,b+c),$$ and if$$N=\left\{\begin{bmatrix}x&-y\\y&x\end{bmatrix}\,\middle|\,x,y\in\Bbb R\right\},$$then, for each $M\in M_2(\Bbb R)$, $x(M)\in\Bbb R^2\times\{(0,0)\}$ if and only if $M\in N$. Therefore, $N$ is a submanifold of $M_2(\Bbb R)$.