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I have two maps from $SU(2) \to SO(3)$.

For the first map, think of $SU(2)$ as the group of unit quaternions. Under this identification, we can give a map $f: SU(2) \to SO(3)$ given by $SU(2)$ acting via conjugation on the 3-dimensional vector subspace of imaginary quaternions with basis $\{i,j,k\}$.

Now think of $SU(2)$ as a group of linear fractional transformations, i.e., to the matrix $(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix})$, we associate the LFT $f(z) = \frac{az + b}{cz + d}$. This map $f: \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ of the Riemann sphere can be converted to a map of an actual sphere $S^2$ using stereographic projection onto the equatorial plane, where it can be extended to a rotation in $\mathbb{R}^3$, giving another map $g: SU(2) \to SO(3)$.

Depending on your conventions, these two maps may not be identical, but they should differ by a change of basis. In fact, for most standard conventions, the change of basis will be a permutation up to signs.

This result can be shown through direct but painful and unenlightening computations, which was what I did. So my question is:

Is there is an easier (more intuitive, more visual, etc.) way to see this?

I'm asking this because it is not too hard to intuitively "see" these rotations, so there might be a way to "see" that these rotations are the same (up to a change of basis).

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Well, the question is, more or less, how to construct a natural action of the group of unit quaternions on a sphere by Moebius transformations. (And related question: what extension of $Sp(1)$ does correspond to the whole group of Moebius transformations?) The answer is... one should look at the night sky.

Let's start with a 4-dimensional vector space with a metric of signature $(1,3)$. The group $SO(1,3)$ acts on the «celestial sphere» (the space of null-lines through the origin) $S^2$ by Moebius transformations. Actually, the induced map $SO^+(1,3)\to PSL_2(\mathbb C)$ is an isomorphism.

Now $\mathbb H$ has a natural scalar product of signature $(1,3)$ — namely, $(u,v)=\operatorname{Re}(u\bar v)$. So the action of $Sp(1)$ on $\mathbb H$ by inner automorphisms preserves this product and we get a map $Sp(1)\to SO^+(1,3)$. Now, on one hand, this action preserves pure imaginary quaternions — i.e. it's image lies in $SO(3)$ (and this is your first map $SU(2)\to SO(3)$ precisely); on the other hand, the image of $Sp(1)$ in $PSL_2(\mathbb C)$ is your second $SU(2)$.

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