Can we define the group SU(2) without using matrices?

Question

If instead in terms of matrices, the group U(2) is defined as the set of all unitary operators in two-dimensional complex vector space that leaves the norm (or norm square) of vectors, $$||\vec{v}||^2=(\vec{v},\vec{v}),$$ unchanged, how will the group SU(2) be defined in this language, without matrices?

Answer 1

You can deine $SU(2)$ by the quaternions $\mathbb{H}$, by taking the numbers $z= a+bi+cj+dk$ such that $a^2+b^2+c^2+d^2=1$. This set with standard quaternion multiplication turns out to be a group, namely $SU(2)$. This is actually a construction that can be much more generalised, and we get the $Spin(p,q)$ groups and Clifford Algebras $Cl(p,q)$. In this case, we constructed $\mathbb{H}=Cl(0,3)$ and $SU(2)=Spin(0,3)$, where the $3$ indicates, that we have three independent complex units:$i^2=j^2=k^2=-1$.