Do anybody know , why can we write the following decompositions for exceptional Lie algebras  $E_6$, $E_7$ and $E_8$
 
1) $E_8=(V^{\star}\otimes \wedge^{8}V^{\star})\bigoplus \wedge^{6}V^{\star}\bigoplus \wedge^{3} V^{\star} \bigoplus \mathfrak{g}\mathfrak{l}(V)\bigoplus  \wedge^{3} V\bigoplus \wedge^{6} V \bigoplus(V\otimes \wedge^{8}V)$, here $dimV=8$

2)  $E_7= \wedge^{6}V^{\star}\bigoplus \wedge^{3} V^{\star} \bigoplus \mathfrak{g}\mathfrak{l}(V)\bigoplus  \wedge^{3} V\bigoplus \wedge^{6} V $,here $dimV=7$

3) $E_6= \wedge^{6}V^{\star}\bigoplus \wedge^{3} V^{\star} \bigoplus \mathfrak{g}\mathfrak{l}(V)\bigoplus  \wedge^{3} V\bigoplus \wedge^{6} V $,here $dimV=6$