In my lecture notes, it's given that
The three-dimensional Heisenberg algebra is a Lie algebra whose three basis elements $p,q,c$ satisfy $[p,q]=c$ and $[c,p]=[c,q]=0$
How would one formally check that this is a Lie algebra? I know we need to check that $[x,x]=0$ and that the Jacobi identity holds for all $x$ in the Lie algebra (though we only need to check it holds for the basis elements $p,q,c$). However I don't see from just the above, how we can check this. Why is $[c,c]=0$ for instance?
The convention is to list only nonzero Lie brackets, and to omit the ones implied by skew-symmetry. So the Heisenberg Lie algebra $L$ with basis $(p,q,c)$ is defined by the bracket $[p,q]=c$. By definition, $[c,c]=0$. Since $[L,[L,L]]=0$, the Jacobi identity amounts to $0+0+0=0$.