How is the exceptional 14-dimensional Lie algebra $\mathfrak{g}_2$ related to the covariant algebra for the binary cubic?

Here are some details on this question. This algebra is generated by 4 forms, these being the form itself $Q$, the discriminant $\Delta$, the Hessian $H$, and a transvectant $T$ of $Q$ and $H$. These four forms ${Q,H,T,\Delta}$ obey a syzygy because $Q^2\Delta$ is a linear combination of $H^3$ and $T^2$. (This is well-known. One may find an exposition in Peter Olver's book on classical invariant theory, for example.)

Let $(d,w)$ denote the degree and weight of a form. Then the degrees and weights of ${1,Q,H,T,\Delta}$ are respectively ${(0,0),(1,3),(2,2),(3,3),(4,0)}$. This sequence is peculiar because these nearly give the branching weights of the adjoint representation of $\mathfrak{g}_2$ with respect to a Lie subalgebra generated by a pair of opposite short root vectors. The only one "missing" is a pair of the form $(2,0)$, which cannot occur.

Is this just a happy coincidence, or is there some "deep" reason reason for this?