Suppose we have two manifolds $M,N$ with Lie brackets
$\chi(M)\times\chi(M)\ni (X,Y) \longmapsto [X,Y]_M \in \chi(M)$ and $\chi(N)\times\chi(N)\ni (X,Y) \longmapsto [X,Y]_N \in \chi(N)$.
My question is what is the relation between $[\cdot,\cdot]_M$, $[\cdot,\cdot]_N$ and Lie bracket $[\cdot,\cdot]_{M \times N}:=[\cdot,\cdot]$ on $M\times N$? Especially is it true that $[(X,0),(Y,0)]=([X,Y]_M,0)$ or $[(X,0),(0,Y)]=0$ or perhaps there is a formula for $[(X,Y),(Z,T)]$? I have tried to google but without any result, and trying to compute looks too complicated for me.
Edit: Actually I would be gratefoul if someone gave at least an answere because I realised that for example in formula: $$[(X,0),(Y,0)]=([X,Y]_M,0)$$ one has a problem even with such an issue that $X,Y$ are not a vector fields on $M$. So are there any relation for those brackets?
Answere:(for decendants) Generally there is no nice answere, yet if $X,Z$ are lifts of vector fields $X,Z$ on $M$ and the same for $Y,T$ than $$[(X,Y),(Z,T)]=([X,Z]_M,[Y,T]_N)$$