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

$$(*)\qquad \{v\in T_xP\mid \omega(v)\in C(\mathfrak{g}),\quad x\in P\}$$

is integrable.

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.