Let $(M,g)$ be a Riemannian manifold. By a theorem of Myers and Steenrod, the group of isometries $Iso(M)$ is a Lie group. By the closed subgroup theorem, the closure of any subgroup of isometries is also a Lie group. I'm curious to understand how well behaved this closure is.
More specifically, suppose $X_1, \ldots, X_n$ are Killing vector fields on $M$ and take $G$ to be the closure of the group of isometries that the flows of $X_i$ generate. How can we describe the invariant/basic forms with respect to $G$? Is it true that if $Lie_{X_i}\alpha = 0$ for $i$ from $1$ to $n$, then $\alpha$ is invariant with respect to $G$? If true, could you provide a reference for a proof? If false, are there additional conditions on the vector fields or on the manifold/metric that would ensure that $Lie_{X_i} \alpha = 0$ is enough for invariance under $G$?
(The same questions could be asked for horizontal forms, defined as $\iota_{X_i} \alpha = 0$)