Tell me more ×  Please give suggestions about soft to make
symbolic computations with NON-commutative variables.  Typical
examples I am interesting - Capelli identities's_identity  For
example let 2x2 matrix X be defined:  $(x_{11}~~~ x_{12})$
$(x_{21}~~~ x_{22})$  and D is defined:  $(\partial_{x_{11}}~~~
\partial_{x_{12}})$  $(\partial_{x_{21}}~~~ \partial_{x_{22}})$
Then there is identity: $det^{column}(XD^t+diag(1,0)) =
det(X)det(D)$  Question Is there soft which can easily check it
?  The ideal would be if some one can provide example of code
checking this thing. (I mean it is true, of course, just to
understand how code such things).  More general things I would
like to do - some computations in universal enveloping of Lie
algebras - like check two expressions commute. e.g. check that
ef+h^2+h is Casimir for sl(2).  I am familiar with MatLab and
Mathematica - but it seems they cannot do this. May be I am
wrong ?  I know that MatLab can differentiate Diff( p,x) - will
give symbolic derivative of symbol