Let $\Delta^+$ denote a choice of positive roots for the Lie algebra $\mathfrak{su}(n)$.
The Weyl vector, $\rho$, is half the sum of the positive roots \begin{equation} 2\rho = \sum_{\beta \in \Delta^+} \beta \,. \end{equation}
Similarly, the Cartan matrix, $A$, can be written as \begin{equation} nA = \sum_{\beta \in \Delta^+} \beta \otimes \beta \,. \end{equation}
What is known about the general case: \begin{equation} C_K = \sum_{\beta \in \Delta^+} \underbrace{\beta \otimes \cdots \otimes \beta}_K \, ? \end{equation} Are the $C_K$ related to other, perhaps better known, objects? Particular $K$-tensors in $\mathfrak{su}(n)$? I am especially interested in the case where $K = 3$.