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It is true in types $G_2$, $F_4$ and $E_6$, but in types $E_7$ and $E_8$ one has to exclude $p=7$, $p=13$ and $p=19$ as well (I think $p=19$ occurs only in type $E_8$). More on this can be found in Jantzen's paper "First cohomology groups for classical Lie algebras" published in Progress in Math., vol. 95, 1991 (the best way to find this paper is to google the title). For related results, see also the paper by Gilkey and Seitz titled "Some representations of exceptional Lie lagebrasalgebras" and published in Geom. Dedicata, 25, (1988), 407-416.

That the Weyl module $V(\varpi_6)$ for $G=E_7(K)$ is reducible in characteristic $7$ is quite easy to see directly: the minuscule module $V(\varpi_7)$ has a nondegenerate $G$-invariant symplectic form and by using Weyl's formula one observes that $V(\varpi_6)$ can be obtained from the second fundamental Weyl module for $Sp_{56}(K)$ by restriction to $G$. The latter Weyl module is reducible in characteristic $7$ as $7$ divides $56$.

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It is true in types $G_2$, $F_4$ and $E_6$, but in types $E_7$ and $E_8$ one has to exclude $p=7$, $p=13$ and $p=19$ as well (I think $p=19$ occurs only in type $E_8$). More on this can be found in Jantzen's paper "First cohomology groups for classical Lie algebras" published in a Progress in Math., vol. 95, 1991 (the best way to find this paper is to google the title). For related results, see also the paper by Gilkey and Seitz titled "Some representations of exceptional Lie lagebras" and published in Geom. Dedicata, 25, (1988), 407-416.

That the Weyl module $V(\omega_6)$ V(\varpi_6)$ for $G=E_7(K)$ is reducible in characteristic $7$ is quite easy to see directly: the minuscule module $V(\omega_7)$ V(\varpi_7)$ has a nonzero nondegenerate $G$-invariant symplectic form and by using Weyl's formula one observes that $V(\omega_6)$ V(\varpi_6)$ can be obtained from the second fundamental Weyl module for $Sp_{56}$ Sp_{56}(K)$ by restriction . to $G$. The latter Weyl module is reducible in characteristic $7$ as $7$ divides $56$.

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It is true in types $G_2$, $F_4$ and $E_6$, but in types $E_7$ and $E_8$ one has to exclude $p=7$, $p=13$ and $p=19$ as well (I think $p=19$ occurs only in type $E_8$). More on this can be found in Jantzen's paper "First cohomology groups for classical Lie algebras" published in a Progress in Math., vol. 95, 1991 (the best way to find this paper is to google the title). For related results, see also the paper by Gilkey and Seitz titled "Some representations of exceptional Lie lagebras" and published in Geom. Dedicata, 25, (1988), 407-416.

That the Weyl module $V(\omega_6)$ for $G=E_7(K)$ is reducible in characteristic $7$ is quite easy to see directly: the minuscule module $V(\omega_7)$ has a nonzero $G$-invariant symplectic form and by using Weyl's formula one observes that $V(\omega_6)$ can be obtained from the second fundamental Weyl module for $Sp_{56}$ by restriction. The latter Weyl module is reducible in characteristic $7$ as $7$ divides $56$.