I am trying to reconcile the notions of algebraic groups, linear algebraic groups, Lie groups, and Lie algebras, along with their notions of root systems, maximal tori, etc. To begin, I am trying to draw a sort of Venn diagram relating algebraic groups, linear algebraic groups, and Lie groups. Certainly all linear algebraic groups are algebraic groups. I know that the universal cover of $SL(2, \mathbb{R})$ is an example of a Lie group which is not a linear algebraic group.
Are all linear algebraic groups Lie groups? I know that they can all be realized as subgroups of some $GL(n, k)$, which is a Lie group, but I believe that a subgroup of a Lie group must be closed to be a Lie group itself.
Thanks for the help!
This is mentioned in the comments, but it is my belief that answers to questions should be in the answer section.
If you look at the Wikipedia article on linear algebraic groups you'll see they are defined to be (in different terms than they use, but better suited to the discussion) Zariski closed subgroups of the group of invertible $n\times n$ matrices. Over the real or complex numbers, this implies that they are closed in the usual topology on this Lie group. Being a closed subgroup of a Lie group, such a group is itself a Lie group. So all linear algebraic groups are Lie groups over a field for which the general linear group is a Lie group. In fields where the general linear group is not a Lie group (for example over $\mathbb Q$) a linear algebraic group need not be Lie group, so there is an example. However it still may be, for example if it has the discrete topology.