On $Q$-polynomial distance-regular graphs with a linear dependency involving a $3$-clique

Let $\mathrm{\Gamma }$ denote a distance-regular graph with diameter $D\ge 2$. Let $E$ denote a primitive idempotent of $\mathrm{\Gamma }$ with respect to which $\mathrm{\Gamma }$ is $Q$-polynomial. Assume that there exists a $3$-clique $\{x,y,z\}$ such that $E\widehat{x},E\widehat{y},E\widehat{z}$ are linearly dependent. In this paper, we classify all the $Q$-polynomial distance-regular graphs $\mathrm{\Gamma }$ with the above property. We describe these graphs from multiple points of view.