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I learnt a while ago about how finite groups of a given dimension can often be broken down using Sylow's theorems into direct/semidirect products of simple groups. I am now studying Lie groups and have learnt that the classification of simple Lie groups is known. How might I go about working out the details for determining simple REAL Lie groups of dimension 1,2,3 etc?

For dimension 1: I have learnt that the connected Abelian Lie groups are of the form $\mathbb{T}^k\times\mathbb{R}^{n-k}$ for some $0\leq k\leq n$. So the connected Abelian Lie groups of dimension 1 are just $\mathbb{R}$ and $S^1$. The only real Lie algebra of dimension 1 is $\mathbb{R}$ with trivial Lie bracket $[x,y]=0$. How can I go about proving that there are no connected, non-Abelian Lie groups of dimension 1?

How could I find all simple Lie groups for dimension 2,3,...? (or to reduce/simplify in whatever way is possible)

Thanks for your help

user829347
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    For simple Lie algebras over $\mathbb R$, see https://math.stackexchange.com/q/3121110/96384. These are in 1-1 correspondence with the simply-connected, connected Lie groups. Further, you can undo the "simply connected" and "connected" restriction via finite extensions / quotients. In particular, the first possible dimensions for simple real Lie groups are $3$ (e.g. $SU(2)$ and $SL(2, \mathbb R)$) and $6$. – Torsten Schoeneberg Jul 27 '20 at 08:00

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