Prove that actions commute

Question

I am trying to understand a proof from Kobayashi and Nomizu (foundations of differential geometry, p. 280). Suppose that we have Lie subalgebras $a<b<g$, with $g$ the Lie subalgebra of $SO(n)$ and $a$ and $b$ Lie algebras of connected Lie subgroups $A$ and $B$ respectively and $B$ is the closure of $A$. The lie group generated by $a$ acts irreducibly on a certain $V<\mathbb{R}^n$. The inner product on $g$ is the negative of the Killing form. Let $a^*$ be the orthocomplement of $a$ in $b$. Let $A_V$ be the action of $A$ on $V$. Suppose we have: $$A\in a^*$$ $$B\in a$$ They now conclude that $A_V$ and $B_V$ commute. How do they conclude this? I figured that $$K(A,B)=0$$ $$tr(A_V)=tr(B_V)=0$$ But I am stuck.