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I know how to prove that $o(4)$ is not a simple Lie algebra, which is finding explicitly two subalgebras that commute with each other. I can show for the Lorentz group that the analogous combination of the infinitesimal generators doesn't give us two invariant subalgebras. Of course, this doesn't prove that the Lorentz group is simple.

I have not found any useful theorem about the conditions for a Lie group to be simple (as the Sylow theorem for finite groups). So far I see the resolution can only be based on the definition of a simple group:

Let G be a group of $r$ parameters. Let $X_{\rho}$ be the generators of its Lie algebra. Let the structure constant $c_{\kappa\lambda}^{\rho}$ be defined as $[X_{\kappa},X_{\lambda}] = c_{\kappa\lambda}^{\rho} X_{\rho}$ (using Einstein notation).

If there is no way to choose the basis elements $X_{\rho}$ of the algebra such that the structure constant $c_{\kappa\lambda}^{\rho}=0$ for $\kappa=1,\ldots,p$ and $\rho=p+1,\ldots,r$ $\Rightarrow$ G is simple.

Does anyone have an idea on how to approach this problem?

PhysicsDevote
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  • Would [math.se] be a better home for this question? – Qmechanic Feb 07 '19 at 01:29
  • Proving by hand that a given Lie algebra is simple can be quite cumbersome. Are you aware of (the Cartan-Killing classification of simple complex Lie algebras, and) Cartan's classification of the real simple Lie algebras? In particular, all $so(p,q)$ with $p+q \ge 3$ are simple. Modern treatments for this are books by Helgason (Differential Geometry, Lie Groups, and Symmetric Spaces), Knapp (Lie Groups Beyond an Introduction), and Onishchik & Vinberg (Lie Groups and Lie Algebras). – Torsten Schoeneberg Feb 08 '19 at 06:03
  • Correction to previous comment (thanks @Rufus): In fact, $so(4,0) \simeq su(2) \oplus su(2)$ and $so(2,2) \simeq sl(2) \oplus sl(2)$ are only semisimple, not simple. To see that $so(3,1)$ is simple, e.g. use it is isomorphic (but as a real Lie algebra, of dimension 6) to $\mathfrak{sl}_2(\mathbb C)$ (viewed as a real Lie algebra of dimension 6, "forgetting" that it is also a complex Lei algebra). Cf. https://math.stackexchange.com/q/1788671/96384 or https://math.stackexchange.com/q/639749/96384. Scalar restrictions of simple LAs stay simple: https://math.stackexchange.com/q/3742748/96384. – Torsten Schoeneberg Nov 22 '22 at 18:40

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