For a Lie group over $\mathbb{C}$ which is simple, in the sense of no normal subgroups closed in the Zariski topology except for finite central subgroups and the whole thing, I'm trying to refresh my memory of how to show that in that case your Lie algebra over C is going to be simple in the sense of no ideals except for zero and the whole thing.

We have a correspondence between subalgebras and subgroups, and any subalgebra of your Lie algebra will give rise to a Lie group and closed immersion of that into your original Lie group, but I would presume that there will be no guarantee the range will be Zariski closed. So I'm just trying to remind myself how to fill in the rest of the argument. Is this result stated in my first paragraph actually correct for $\mathbb{C}$?

My main reason for asking this is that I'm wondering whether it can be generalised to non-archimedean local fields, but I thought it might be convenient to start by thinking about the case of the complex numbers.

For an affine algebraic group over $\mathbb{C}$ which is simple, in the sense of no normal subgroups closed in the Zariski topology except for finite central subgroups and the whole thing, I'm trying to refresh my memory of how to show that in that case your Lie algebra over C is going to be simple in the sense of no ideals except for zero and the whole thing.

We have a correspondence between subalgebras and subgroups, and any subalgebra of your Lie algebra will give rise to a Lie group and closed immersion of that (relative to the strong topology) into your original Lie group, but I would presume that there will be no guarantee the range will be Zariski closed. So I'm just trying to remind myself how to fill in the rest of the argument. Is this result stated in my first paragraph actually correct for $\mathbb{C}$?

My main reason for asking this is that I'm wondering whether it can be generalised to non-archimedean local fields, but I thought it might be convenient to start by thinking about the case of the complex numbers.