How can I prove that every complex lie group contains a two-dimensional subgroup. I'm trying to solve this question in the book "applications of lie groups to differential equations by Olver"
Prove that every complex lie group contains a two-dimensional subgroup. Is it true for real lie groups? Consider the vector V in the lie algebra. The image of C.v under the exponential map from g to G is supposed to be that specific two dimensional subgroup. Can someone tell me how? And why isn't true for real lie groups?
 A: After looking through Olver's book, here is the story.

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*He is sloppy and occasionally forgets to give definitions, forgets to assume some conditions, etc. Be aware of this when reading. For instance, in the very next exercise 1.24, in part (b) he forgot to assume that $G$ is connected, in part (c) he forgot to assume that $H$ is a closed subgroup and in part (e) he forgot to say "proper," which means "nontrivial and different from $SO(3)$. He also probably forgot to assume that the normal subgroup is a Lie subgroup: Without this assumption, the same conclusion (i.e. $G$ is simple as an abstract group) still holds but requires a very different argument, using tricks not covered in Olver's book.


*He've made it very hard to solve Exercise 1.23 because he forgot to define what a complex Lie group is and forgot to explain what the notion of dimension refers to. Moreover, he forgot to explain that all the results about real Lie groups also apply to complex Lie groups. Lastly, he forgot to assume that $G$ has positive dimension.


*I am not going to go through the entire theory, but a complex Lie group $G$ is a complex manifold, equipped with group operations which are differentiable in the sense of complex analysis. In particular, the Lie algebra of a complex Lie group is both real and complex Lie algebra. In particular, 1-dimensional complex linear subspaces of the Lie algebra are subalgebras. It follows that if the complex Lie algebra ${\mathfrak g}$ of $G$ has positive complex dimension (which is always the case provided that $G$ has positive dimension), then it contains 1-dimensional complex subalgebras ${\mathfrak a}$, which are then real 2-dimensional (commutative) subalgebras. Thus, Theorem 1.51 applies to ${\mathfrak a}$ and $H=\exp({\mathfrak a})$ is a Lie subgroup of the group $G$, whose Lie algebra is isomorphic to ${\mathfrak a}$. Note that Olver does not require submanifolds to be embedded, only injectively immersed. Hence, Lie subgroups in his sense need not be closed. All in all, we obtained a 2-dimensional (or complex 1-dimensional) Lie subgroup $H$ of $G$.
