There is another approach by the strange formula of Freudenthal and de Vries, which states that $h^\vee d = 12 \rho^2$ where the Weyl vector $\rho$ is one half of the sum of positive roots. The long roots have the length square two. Thus $$\frac13 h^\vee d = \sum_{\alpha>0} \alpha^2+ 2\sum_{\alpha,\beta>0,\alpha\neq\beta}\alpha\cdot\beta \sum_{\alpha,\beta>0,\alpha\neq\beta}\alpha\cdot\beta$$ For simple-laced group, the first term of RHS is the number of roots $hr=d-r$, and the second term of RHS should be $rh(h-2)/3$. For the non-simple-laced case, there may be still some interesting interpretation of the above decomposition.
There is another approach by the strange formula of Freudenthal and de Vries, which states that $h^\vee d = 12 \rho^2$ where the Weyl vector $\rho$ is one half of the sum of positive roots. The long roots have the length square two. Thus $$\frac13 h^\vee d = \sum_{\alpha>0} \alpha^2+ 2\sum_{\alpha,\beta>0,\alpha\neq\beta}\alpha\cdot\beta$$ For simple-laced group, the first term of RHS is the number of roots $hr=d-r$, and the second term of RHS should be $rh(h-2)/3$. For the non-simple-laced case, there may be still some interesting interpretation of the above decomposition.
There is another approach by the strange formula of Freudenthal and de Vries, which states that $h^\vee d = 12 \rho^2$ where the Weyl vector $\rho$ is one half of the sum of positive roots. The long roots have the length square two. Thus $$\frac13 h^\vee d = \sum_{\alpha>0} \alpha^2+ 2\sum_{\alpha,\beta>0,\alpha\neq\beta}\alpha\cdot\beta$$ For simple-laced group, the first term is the number of roots $hr=d-r$, and the second term should be $rh(h-2)/3$. For the non-simple-laced case, there may be still some interesting interpretation of the above decomposition.