Are the Complex Numbers a subset of the Split Quaternions?

Question

I saw that if $a+bi\in \mathbb C$, it's also an element of the split quaternions ($\mathbb P$), since $a+bi=a+bi+0j+0k$. Does this mean $\mathbb C\subset\mathbb P$? If so, does it follow that all Cayley-Dickenson Constructions are a subset of the split version of twice the dimensions?

I remember from set theory that if $(x \in A)\rightarrow(x\in B)$ then $A\subset B$, but I don't know if it applies here, since $A$ and $B$ might have different properties.

Answer 1

It's cleaner to say that there's a natural embedding $\Bbb C\hookrightarrow \Bbb P$ of the ring (field) of complex numbers to the ring of split quaternions, making $\Bbb C$ isomorphic to a subring of $\Bbb P$.

Note that this is the general pattern in any construction step of numbers: $$\Bbb N\ \hookrightarrow\ \Bbb Z\ \hookrightarrow\ \Bbb Q\ \hookrightarrow\ \Bbb R\ \hookrightarrow\ \Bbb C$$ In particular, this last step can be introduced by defining $\Bbb C:=\Bbb R[x]/(x^2+1)$, and then we can identify elements of $\Bbb R$ with the cosets of constant polynomials, thus defining an injective homomorphism $\Bbb R\to\Bbb C$, but we have no mere inclusion $\Bbb R\subseteq\Bbb C$.

Answer 2

Split-quaternions can be represented as arbitrary real $2\times2$ matrices $\left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right)$. On the other hand, complex numbers can be represented by $2\times2$ matrices of the following form: $\left( \begin{array}{cc} a & b \\ -b & a \\ \end{array} \right)$, so they are isomorphic to a subset of split-quaternions.