See Exercise 6 in Page 49 of the following book of E. I. Khukhro:
$p$-Automorphisms of Finite $p$-Groups, London Mathematical Society Lecture Note Series, 246, Cambridge University Press, Cambridge, 1998.

- Let $G$ be a group generated by two elements $x$ and $y$. Show that the law $\delta_2$
of solubility of derived length 2 holds on the generators $x, y$ (while $G$ may
not be soluble: for example, $\mathbb{S}_5$ is generated by a cycle of order $5$ and a
transposition).

The law $\delta_2$ is $[[x_1,x_2],[x_3,x_4]]$.