Exercise 2, § 13

Let $(\mathfrak{g},\mathfrak{h})$ be a split simple Lie algebra, and B a basis of $R(\mathfrak{g},\mathfrak{h})$. The simple $\mathfrak{g}$-modules $E(\lambda ),\lambda \in P_{++}$ -- $\{0\}$, of minimum dimension are those for which $\lambda$ is one of the following weights:

$\varpi_1(A_1);\varpi_1$ and $\varpi_l(A_l,l\geq 2);\varpi_1(B_l$ and $C_l,l\geq 2);\varpi_1, \varpi_3$ and $\varpi_4$ $(D_4);\varpi_1(D_l,l\geq 5);\varpi_1$ and $\varpi_6(E_6);\varpi_7(E_7);\varpi_8(E_8);\varpi_4(F_4);\varpi_1(G_2)$.

Any two such modules can be transformed into each other by an automorphism of $\mathfrak{g}$.

Type $E_8$ is the only one for which the adjoint representation is of minimum dimension.