I have some question about detail on Chapter 9 Riemannian submersion in the book [Be].

[Be] A. L. Besse, Einstein Manifolds, Springer-Verlag

In the above book, in the proof of 9.24 Proposition (See p. 240) it is described that $[X,U]$ is vertical.

Let me explain details : On $(M = B\times_f F, g=\hat{g} + f\check{g})$ with a

submersion $\pi : M \rightarrow B$, let $U$ be a vertical vector field and $X$
be a horizontal vector field.

The proof used the argument that $[X,U]$ is vertical, which is not proved. I can not understand why $[X,U]$ is vertical.

In pp. 239-240, 9.22 also says that $[X,U]$ is vertical, but no explanation. And I know that $d\pi[X,Y] = [d\pi X, d\pi Y]$. From this can we derive that $[X,U]$ is vertical ?

Please give me an explanation