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A Lie group $G$ naturally acts on its Lie algebra $\gg$, called the adjointaction. In this paper, we determine the orbit types of the compact exceptionalLie group $G_2$ in its Lie algebra $\gg_2$. As results, the group $G_2$ hasfour orbit types in the Lie algebra $\gg_2$ as $$ G_2/G_2, \quad G_2/(U(1)\times U(1)), \quad G_2/((Sp(1)\times U(1))/\Z_2), \quad G_2/((U(1)\timesSp(1))/\Z_2). $$ These orbits, especially the last two orbits, are notequivalent, that is, there exists no $G_2$-equivariant homeomorphism amongthem.

Adjoint orbit types of compact exceptional Lie group G2 in its Lie algebra