Constructing a basis (and structure constants) from a list of generators

Question

I have $2k$ generators given by $x_{\alpha}$ and $y_{\alpha}$ (where $\alpha$ ranges from $1$ to $k$) that satisfy the following commutation relations: $$[y_{\alpha}, x_{\beta}] = (d+2w) \delta_{\alpha \beta} + \tfrac{2}{d+2w-2} (x_{\beta} y_{\alpha} - x_{\alpha} y_{\beta})\,,$$ $$[x_{\alpha}, x_{\beta}] = 0\,,$$ where $d$ and $w$ can be treated as arbitrary real numbers (i.e. don't worry about dividing by zero) and $\delta_{\alpha \beta}$ is the usual Kronecker delta. Note that the commutator $[y_{\alpha},y_{\beta}]$ would presumably have to be a new basis element because it cannot be expressed in terms of just $x_{\alpha}$ and $y_{\alpha}$ besides just the commutator itself.

Were $k=1$, I would be able to identify these generators as two (of three) basis elements for an $sl(2)$ algebra, but the non-linearities for $k \neq 1$ make it much harder to understand the algebra that this set of generators are part of.

I am hoping one of you wizards can help me answer the following questions.

1. Do these generators and commutation relations uniquely generate a "minimally-generated" algebra?
2. If so, is there a straightforward way to see if this minimally-generated algebra is finite dimensional?
3. If so, is there an algorithmic way to generate a basis for this minimally-generated algebra?
4. Do these generators and commutation relations generate a well-known algebra?
5. Is there a way to tell if these generators and commutation relations generate a Lie algebra without knowing the commutation relations for every pair of basis elements?

My initial thought was simply to just try defining a basis element for every new (non-linear) term that arose in the commutator, so that I might define $z_{\alpha \beta} = x_{\alpha} y_{\beta}$, and then iteratively compute commutators from there. However, it was not clear that this procedure would terminate and "close up." And before I go and try to actually do this, it would be great to know if it is indeed possible to show that it'll close, etc.

(Be kind, I'm just a physicist working in differential geometry!)