I am interested to expand the symmetrised tensor products of several elements of the Philip Hall basis of a free Lie algebra in tensor form. For example, if the algebra has two generators $x$ and $y$, then we have $\ell_1=x$, $\ell_2=y$, $\ell_3=[x,y]$, and so on. It is easy to calculate by hand that $$ (\ell_1,\ell_2,\ell_2)=\frac{1}{3}(x\otimes y\otimes y+y\otimes x\otimes y+y\otimes x\otimes x), $$ but the calculation of,say, $(\ell_1,\ell_2,\ell_5)$ is more boring. Do there exist a software which may do such calculations?
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1Note, Magma has a command FreeLieAlgebra but, the last time I checked, this does not -- and is not meant to -- produce a Lie algebra. This is an important non-example! – David A. Craven Dec 27 '22 at 14:27
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GAP may be useful here, but I've never used it's Lie algebra features https://docs.gap-system.org/doc/ref/chap64.html – diracdeltafunk Dec 27 '22 at 15:31
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1I tried GAP. It successfully created free Lie algebra $L$ with two generators but failed to create its universal enveloping algebra. – Anatoliy Malyarenko Dec 27 '22 at 20:10
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I found that the web page https://sagecell.sagemath.org/ does calculations I require. – Anatoliy Malyarenko Jan 04 '23 at 18:08