Currently struggling with showing that the quaternions are isomorphic to a subgroup of $M_2(\mathbb{C})$.
I've defined a map $\phi$ from $\mathbb{R}\langle x,y,z\rangle$ to $M_2(\mathbb{C})$ such that $x\mapsto \begin{pmatrix}i&0\\0&-i\end{pmatrix}$, $y\mapsto \begin{pmatrix}0&1\\-1&0\end{pmatrix}$ and $z\mapsto \begin{pmatrix}0&i\\i&0\end{pmatrix}$.
I've shown that this is a homomorphism, and that the ideal $I=(x^2+1,y^2+1,z^2+1,xyz+1)$ of $\mathbb{R}\langle x,y,z\rangle$ is contained in the kernel of $\phi $, but cannot show the other inclusion.
My method has relied on supposing that an element exists in $ker\phi\setminus I$, taking its image and trying to show that it must be multiples of $f(x)^2+1$, etc. This has not worked, and I have no idea how else to go about it.
I've had a look at some of the other questions similar to this, but they usually involve ideals with only one generator. I was wondering if there was a more simple method to showing the reverse inclusion.