For a classical group $ \mathrm{SU}(n),\mathrm{SO}(n),\mathrm{Sp}(n) $ there is a nice way to construct the adjoint representation from the natural module $ V $. For $ \mathrm{SU}(n) $ the adjoint is $ (V^* \otimes V) -1 $, for $ \mathrm{SO}(n) $ the adjoint is $ \Lambda^2(V) $, and for $ \mathrm{Sp}(n) $ the adjoint is $ S^2(V) $. For details see Decomposition of $ V \otimes V^* $ for the natural representation.
In this way we got a relationship between the minimal dimensional irrep of the classical group, and the adjoint rep of the classical group.
What about exceptional groups? Is there a nice way to relate the adjoint rep of the exceptional groups $ G_2,F_4,E_6,E_7,E_8 $ to the minimal rep?
I know the minimal irreps are \begin{align*} G_2 &\hookrightarrow \mathrm{SO}(7) \\ F_4 &\hookrightarrow \mathrm{SO}(26) \\ E_6 &\hookrightarrow \mathrm{SU}(27) \\ E_7 &\hookrightarrow \mathrm{Sp}(28) \\ E_8 &\hookrightarrow \mathrm{SO}(248) \end{align*} For $ E_8 $ there isn't much to be said since the minimal irrep is also the adjoint rep. But for the other four I would expect: the adjoint rep of $ G_2 $ is some suitably defined subrep of $$ \Lambda^2(\textbf{7}) $$ adjoint rep of $ F_4 $ some suitably defined subrep of $$ \Lambda^2(\textbf{26}) $$ adjoint rep of $ E_6 $ some suitably defined subrep of $$ (\textbf{27} \otimes \overline{\textbf{27}})-1 $$ and adjoint rep of $ E_7 $ some suitably defined subrep of $$ S^2(\textbf{56}) $$ Are any of these constructions well-known/ simple/ interesting/ related to other interesting objects like actions on lattices or Albert algebras?
