I am undergrad trying to get into $SU(2)$ invariant instanton geometries, and I am very confused about how to formulate a statement in an invariant way.
I know that a general four-dimensional Riemannian metric with $SU(2)$ isometry locally looks like $$g = d r^2 + h_{ij}(r) \sigma^i \sigma^j \tag{1}$$ where $r$ is some radial coordinate and $\sigma^i$, $i,j=1,2,3$, are left-invariant $SU(2)$ 1-forms. These 1-forms can then be parameterized in coordinate form, if necessary, e.g., $\sigma^3 = d\psi + \cos \theta \,d\phi$ and more complicated expressions for $\sigma^{1},\sigma^{2}$.
I have been working with geometries that have $h_{ij}$ as a diagonal metric, e.g., $$h = a^2(r) (\sigma^1)^2 + b^2(r) (\sigma^2)^2 + c^2(r) (\sigma^3)^2 .$$
However, the diagonality of a metric is a frame dependent choice, e.g., $SO(2)$ rotation of $\sigma^{1,2}$ will introduce $r$-dependent cross-terms!
Is there a frame-invariant way of saying that the metric (1) is diagonal?
P.S. Please correct me if I ask an imprecise question, this is my first question here!