= Solution
Use <Cartan's magic formula>. The <exterior derivative> and <interior product of a differential form> give
$$
d\alpha=-2\,dx\wedge dy,\qquad \iota_X\alpha=-x^2-y^2+z^2,\qquad \iota_Xd\alpha=2x\,dx+2y\,dy.
$$
Thus $d(\iota_X\alpha)=-2x\,dx-2y\,dy+2z\,dz$, and the first two components cancel:
$$
\boxed{\mathcal L_X\alpha=2z\,dz.}
$$
The rotation in the $xy$-plane leaves $y\,dx-x\,dy$ invariant, while the dilation in $z$ doubles the infinitesimal weight of $z\,dz$, providing a geometric check of this <Lie derivative>.
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