Consider the set of all Lie bracket products of elements in V; i.e., V₁ = {[v,w]: v, w in V}. V₁ is necessarily an ideal subalgebra of V because [[v,w],u]=[z,u] is necessarily an element of V₁. But V₁ will not necessarily include all of the elements of V. It may for some Lie algebras but for others it may not.

Having formed V₁ the process may be repeated. Thus V₂={[v,w]: v,w in V₁} and likewise for V₃ and so on. At some point the process may produce the trivial subalgebra of the additive identity element 0 of V; i.e., for some n, V_n={0}. A Lie algebra such that the process described above terminates in the trivial subalgebra {0} is said to be solvable. The ideal subalgebras of a solvable Lie algebra are themselves solvable.

The Radical of a Lie Algebra

There can be ideal subalgebras of a Lie algebra that are solvable but the Lie algebra itself not solvable. For a Lie algebra its radical subalgebra is defined as the maximal solvable ideal subalgebra. It can be proven that for any Lie algebra such a radical subalgebra exists.

The Simpleness of a Lie Algebra

Any nontrivial Lie algebra V necessarily contains at least two ideal algebras; i.e., V itself and the trivial subalgebra {0}. If there are no other ideal subalgebras the Lie algebra is said to be simple.

A Lie algebra V is said to be abelian if its Lie bracket is symmetric; i.e., [u,v] = [v,u] for all u and v in V. An abelian subalgebra S of a Lie algebra V is one for which the Lie bracket is symmetric in S even if it is not symmetric for all of V. If a Lie algebra V contains no abelian ideal subalgebras other than the trivial one of {0} then V is said to be semi-simple.

The Decomposition of a Lie Algebra

Suppose that for a Lie algebra V there are subsets A and B such that any element v of V can be represented as the sum of an element from A and an element from B; i.e., v=a+b where a is in A and b is in B. This relation is denoted as V=A+B. If furthermore the Lie bracket products of the elements of A and B are all equal to the additive identity of V, [a,b]=0 for all a in A and b in B, then V is said to be the direct sum of A and B. Such A and B are subalgebras of V.

If for a Lie algebra V there are two subalgebras A and B such that V=A+B and the Lie bracket products of the elements of A and B are all contained within A then V is said to be the semi-direct sum of A and B.

The Levy Decomposition Theorem for Lie Algebras

The significance of the concepts defined above is that any Lie algebra is isomorphic to the semi-direct sum of its radical and a semi-simple subalgebra. The French mathematician Elie Cartan was able to classify all of the semi-simple Lie algebras. They all fall into four classes except for five special Lie algebras.

(To be continued.)