$G$ a Lie group with a $K$-bi-invariant metric for a closed Lie subgroup $K\subset G$

Question

Let $K$ be a closed (connected if needed) Lie subgroup of $G$. Equip $G$ with a metric that is $G$-left-invariant and $K$-right-invariant. If $X_\mathfrak{k},Y_{\mathfrak{k}}\in\mathfrak{k}\subset\mathfrak{g}$ are such that $[X_{\mathfrak{k}},Y_{\mathfrak{k}}]=0$ then can we say that $X_{\mathfrak{k}}$ and $Y_{\mathfrak{k}}$ are linearlly dependent (as $K$'s inherited metric is bi-invariant and so it has positive sectional curvature)?