I’m considering the rotation group, as described by $SO(3)$ (i.e the set of $3 times 3$ orthogonal matrices $R$ of determinant 1, with real components) and $SU(2)$ (the set of $2 times 2$ complex matrices $U$ of determinant 1). The elements of these two groups could be represented like this (the unit matrix is implied implicitely):

begin{align}

R(alpha, n_k) &= 1 cos alpha + sum_{k = 1}^3 lambda_k , n_k sin alpha + n , n^{top} (1 – cos alpha) quad in SO(3), tag{1} \

U(alpha, n_k) &= e^{i vec{sigma} , cdot , vec{n} , alpha / 2} = 1 cos (alpha/2) + i sum_{k = 1}^3 sigma_k , n_k sin (alpha/2) quad in SU(2), tag{2}

end{align}

where $lambda_k$ ($k = 1, 2, 3$) are three $3 times 3$ skew-symetrical matrices that generate the rotations, and $sigma_k$ are the three $2 times 2$ Pauli matrices. The components $n_k$ are three real numbers defining the rotation axis. We could write these components as this:

$$n_k = { sinvartheta cos varphi, : sin vartheta sin varphi, : cos vartheta }. tag{3}$$

Of course, $alpha$ is the rotation angle.

Now, I define a scalar product in $SU(2)$ as a trace:

$$langle , U_1, , U_2 rangle = frac{1}{2} , mathrm{Tr}(U_1^{dagger} , U_2), qquad U_1, , U_2 , in , SU(2).tag{4}$$

Calculating the differential $dU$ gives us a metric on $SU(2)$ (I set $chi equiv alpha / 2$ to simplify things):

$$ds^2 = frac{1}{2} , mathrm{Tr}(dU^{dagger} , dU) = dchi^2 + sin^2 chi , (dvartheta^2 + sin^2 vartheta , dvarphi^2).tag{5}$$

This is a well known metric on $mathcal{S}_3$, the 3-sphere. This is fine, since $SU(2)$ is a compact Lie group, which have the topology of that sphere.

But then I tried to do the same with $SO(3)$ (the calculations are very messy in this case). I got this:

$$frac{1}{8} , mathrm{Tr}(dR^{top} , dR) = dchi^2 + sin^2 chi , (dvartheta^2 + sin^2 vartheta , dvarphi^2).tag{6}$$

The fraction $frac{1}{8}$ on the left part puzzles me. Why this factor, instead of $frac{1}{3}$? This trace of $3 times 3$ matrices doesn’t look like a scalar product. My interpretation of (6) is lacking. How can we define a proper scalar product and Riemanian metric on $SO(3)$ that would show the 3-sphere hidden in it?