I consider a Lie group $G$, with a group element $g$ parametrised in some manner with parameter $\theta_i$, $i=1,\cdots, \dim G$. Suppose that $K\subset G$. I want to compute the variation of an group element under the left action of $K$ on $G$.
I'm a little confused to how I can do this: Something like $\delta g =g\delta\theta_iT^i$, with $T^i$ the generators of $G$? but restricted to the subgroup $K$?