 Solution

Use Cartan's magic formula. The exterior derivative and interior product of a differential form giveThus $d({\iota }_X\alpha )=-2xdx-2ydy+2zdz$, and the first two components cancel:The rotation in the $xy$-plane leaves $ydx-xdy$ invariant, while the dilation in $z$ doubles the infinitesimal weight of $zdz$, providing a geometric check of this Lie derivative.