Induced Lie algebra homomorphism from mapping unit quaternions to a rotation

Question

Let $U\subseteq\mathbb{H}$ be the set of unit quaternions. We can map a unit quaternion $a\in U$ to map $f_a:\mathbb{R}^3\to\mathbb{R}^3$, given by $f_a(b)=aba^{-1}$, where $b\in\mathbb{R}^3$ is viewed as an imaginary quaternion, $(x,y,z)\mapsto xi+yj+zk$. It turns out that the map $a\mapsto f_a$ is a Lie group homomorphism $U\to SO(3)$. Now I want to find the induced Lie algebra homomorphism from that. How do I do that? Do I have to get hold of a "concrete" description of their Lie algebras to do this, or is there any nicer way?

Answer 1

The Lie algebra of $U$ can be naturally identified with the space of imaginary quaternions via observing that the derivative of $a\bar a=1$ at $a=1$ is given by $b\bar 1+1\bar b=0$, i.e. by $\bar b=-b$. Likewise, the derivative of $f_a$ at $a=1$ in direction $b\in\mathbb R^3$ is given by $c\mapsto bc1^{-1}+1c\bar b$ (using that for $a\in U$ we have $a^{-1}=\bar a$), so this is $c\mapsto bc-cb$. Now you can express this in the basis $i$, $j$, $k$, if you want. The main issue however is that the reuslting map from $\mathbb R^3$ to $\mathfrak{so}(3)$ is injective and thus a linear isomorphism.