Unit quaternion ball is compact and connected?

Question

Let$$\mathbb{U} := \{x \in \mathbb{H} : |x| = 1\}.$$This is a group under multiplication. What is the easiest way to see that $\mathbb{U}$ is a compact and connected subset of $\mathbb{H}\cong \mathbb{R}^4$?

Answer 1

As noted in the comment a unit quaternion $\mathbf{u}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$ is such that $|\mathbf{u}|=a^2+b^2+c^2+d^2=1$

so your set $\mathbb{U}$ is isomorphic tho the unit sphere $S^3$ and this is connected as you can see at How do you prove that the 3-sphere is connected?