Uniqueness of embeddings of the affine line into algebraic groups

Let Y Y be the underlying variety of a complex connected affine algebraic group. We prove that two embeddings of the affine line C \mathbb {C} into Y Y are the same up to an automorphism of Y Y provided that Y Y is not isomorphic to a product of a torus ( C ∗ ) k (\mathbb {C}^\ast )^k and one of the three varieties C 3 \mathbb {C}^3 , SL 2 \operatorname {SL}_2 , and PSL 2 \operatorname {PSL}_2 .