# lie groups – Metric of \$SO(3)\$ : Why is there \$frac{1}{8}\$ in front of this scalar product?

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?