I wanted to know if the structure constants of a Lie group can be function, or if the have to be proper numbers. I think that because of Lie algebra is obtained by derivation of group's elements and evaluation at the identity, they have to be numbers, but is it right? And example of what I mean is:
The vector fields that you can find in the attached image are supposed to form the Lie algebra of the group O(3, 2). It's easily to check that the commutator between $V_1$ and $V_9$ is $[V_1, V_9] = 2tV_6$, what implies an structure constant $f_{196} = 2t$ that is not constant, depends on $t$. But is this possible or the structure constants should be CONSTANTS (proper numbers, not functions)?
Thanks! ;)
Indeed, structure constants of a Lie algebra are constants. They cannot be variables in a specific realization like the one you are proffering. The variables x,t,u of your geometric realization are placeholder props of flows on your realization manifold and should never enter the structure constants of any Lie algebra here. An equivalent set of matrices with identical commutator structure would not "know" a thing about them: they should not be involved.
The freaky commutator you have found is an indication that the 10 Vs you have certainly cannot close into the o(3,2) algebra. You can see that, since this algebra has rank 2, only, but your $V_1=\partial_t,V_4=\partial_x,V_6=\partial_u$ commute among themselves, suggesting rank of at least 3. So you may have miscopied these matrices, or misconstrued what was asserted about them (where, pray tell?).
I suggest studying so(5) and its realizations, and then fiddling with the signs of the structure constants by tweaking the normalizations of the generators. Recall you are studying $B_2\sim C_2$, where $so(3,2;r)\sim sp(4,r)$.
In particular, I am troubled by the so(3) subalgebra structure not being as manifest in your $V_3,V_8, V_9$ as it is on the remaining proposed generators. There appears to be something very wrong. For example, $[V_3,V_9]$ has a cubic polynomial doted on the gradient, not contained in your proposed set. The set is not a complete set of Lie generators.