You're right. This is finite commutation table. Actually it is just array.

How to use it to setup Lie Algebra, using commands from components Lie Algebra and Differential Geometry?

Actualy I want to do some more things with it, like Levi Decomposition and so on...

I'm not sure if you really need reid's script here, because my question only involves how to implement result (array) to standard routines of Lie Algebra, however it is also attached...

stanform.zip

Thank you for really nice answer!

@ecterrab I've made a Maple worksheet with decomposition, but unfortunately it is trivial. What I am doing wrong?P.S: fct.mw

@ecterrab According to example 2 here:

http://www.maplesoft.com/support/help/Maple/view.aspx?path=DifferentialGeometry/LieAlgebras/Decompose

sometimes you have to help Maple with factorization.

@ecterrab: I have a commutation table of Lie algebra. I want to find its maximal abelian dimension - it is just number (like 4 or 5). There are algorithms (even implemented in Maple) to do it. But at first I'm interested if it is available to do it using Maple built-in procedures...

I've looked through. There is a procedure "Decompose", probably it could be useful.

But seems it is based on factorization of polynomial, it is not fully algorithmic...

Eugene Kalentev: Thanks for the reference. But unfortunately I can not find there exactly what i am searching.

Do you have an idea how to find maximal abelian subalgebra using commands from Physics package?

Probably it could be done in some steps, using more advanced techniques...

Let say a - vector functions depends on t.

I want to calculate something like: cross product of a + [a',a] and a' + [a',a], but symbolically (without components).

@Carl Love: thanks a lot!

@ecterrab: thanks a lot!

@Mac Dude: actually, this pretty obvious idea doesn't work in my Maple (18) :(

@Preben Alsholm: Thanks for the answer. Well, actually I'm looking for computing Abstract Lie Symmetry Algebra for ODE without solving determining equation. It seems to me that this feature is not released in Rif in Maple 18.

Thanks for fast and exact answer!

@Carl Love: Thanks!

@Carl Love: Thanks for fast answer. Well, order doesn't matter and repetition of invidual elements doesn't matter too...