Principal bundle, horizontal and vertical subspaces

Question

Given a principal bundle $P(M,G)$, we can decompose $$T_pP=V_pP\oplus H_pP.$$ I don't understand why $$[X,Y]\in H_pP$$ if $X\in H_pP$, and $Y\in V_pP$. Thanks!

Answer 1

The statement is not true. X is element of the tangent base space and Y is element of the tangent fiber space. Because, the fiber direction and the vertical direction are exactly same, that confusion happens. In other words, horizontal space has both fiber and base components. But vertical space only has fiber component.