A property of polarization

Let $G$ be a real Lie group with the Lie algebra $\mathfrak g$, and let f be a real linear functional on $\mathfrak g$. It is established that if $\operatorname{Ker}f$ does not contain nonzero ideals of the algebra $\mathfrak g$, then the existence of a total positive complex polarization for $f$ implies that the Lie algebra of the stationary subgroup of the functional $f$ in $\mathfrak g$ is reductive.