Integrability of certain distribution associated to a connection form on the total space of a principal bundle (Principal Frobenius condition)

Question

Let $P\to M$ be a $G$-principal bundle where $P,M$ are smooth manifolds and $G$ is a Lie group with Lie algebra $\mathfrak{g}$, whose center is denoted by $C(\mathfrak{g})$.

Let $\omega$ be the connection form of a connection for our principal bundle.

We define a distribution on the total space $P$ as follows: $$ \{v\in T_xP\mid \omega(v)\in C(\mathfrak{g}),\quad x\in P\}. \label{1}\tag{$*$}$$ This defines a $G$-invariant distribution on $P$.

Under what algebraic conditions on $\omega$, \eqref{1} is an integrable distribution? What is a precise example of a foliation which can be generated in this way and the Lie algebra $\mathfrak{g}$ is not commutative? Is there an example of this situation such that we have a leaf with non-trivial holonomy? On the other extreme, what is an example of a distribution \eqref{1} which is not integrable?

As a second question, is there a geometric interpretation for the following algebraic condition: $$(\omega \wedge d\omega)(X,Y,Z)\in C(\mathfrak{g}),\quad \forall X,Y,Z\in T_x P,\; x\in P\quad ?$$

Answer 1

For every $u\in C({\cal g})$ there exists a fundamental vector $u^*$ defined on $P$ by $u^*(x)={d\over{dt}}_{t=0}x\exp(tu)$; if $u,v\in C({\cal g})$, $0=[u,v]^*=[u,^*,v^*]$. Frobenius theorem implies the distribution $$ \{v\in T_xP\mid \omega(v)\in C(\mathfrak{g}),\quad x\in P\} \tag{$\ast$} $$ is integrable.