Embedding $SU(2)\times SU(2)\to Cl(\Bbb R^4)$

Question

Consider the spin group $\text{Spin}(4)$. By definition it is contained in the Clifford algebra $Cl(\Bbb R^4)$ (https://en.wikipedia.org/wiki/Spin_group#Construction). Since it is known that $\text{Spin}(n)$ is the universal (double) cover of $SO(n)$ (for $n>2$), $\text{Spin}(4)$ is the universal cover of $SO(4)$. On the other hand, we have a double cover $\pi:SU(2)\times SU(2)\to SO(4)\subset GL(\Bbb R^4)$, defined by $\pi(A,B)(x)=AxB^{-1}~~(x\in \Bbb R^4)$. Thus $\text{Spin}(4)$ must be isomorphic to $SU(2)\times SU(2)$. This means that there is an embedding of $SU(2)\times SU(2)$, given by isomorphism $SU(2)\times SU(2)\to \text{Spin}(4)$ followed by inclusion $\text{Spin}(4)\to Cl(\Bbb R^4)$. What I am curious about is: Can we get an explicit formula of this embedding $SU(2)\times SU(2)\to Cl(\Bbb R^4)$?