Separable forms of {{mathbb{G}}_m} -ACTIONS ON {{mathbb{A}}^3}

Let k be a field of characteristic zero. We consider k-forms of $ {mathbb G} $ _m -actions on $ {mathbb A} $ ³ and show that they are linearizable. In particular, $ {mathbb G} $ _m -actions on $ {mathbb A} $ ³ are linearizable, and k-forms of $ {mathbb A} $ ³ that admit an effective action of an infinite reductive group are trivial.