  Let $\Phi = (\phi_1,\dots,\phi_6)$ be a system of $6$ linear forms in $3$
variables, i.e. $\phi_i \colon \mathbb{Z}^3 \to \mathbb{Z}$ for each $i$.
Suppose also that $\Phi$ has Cauchy--Schwarz complexity $2$ and true complexity
$1$, in the sense defined by Gowers and Wolf; in fact this is true generically
in this setting. Finally let $G = \mathbb{F}_p^n$ for any $p$ prime and $n \ge
1$. Then we show that multilinear averages by $\Phi$ are controlled by the
$U^2$-norm, with a polynomial dependence; i.e. if $f_1,\dots,f_6 \colon G \to
\mathbb{C}$ are functions with $\|f_i\|_{\infty} \le 1$ for each $i$, then for
each $j$, $1 \le j \le 6$:
  \[
  \left| \mathbb{E}_{x_1,x_2,x_3 \in G} f_1(\varphi_1(x_1,x_2,x_3)) \dots
f_6(\phi_6(x_1,x_2,x_3)) \right| \le \|f_j\|_{U^2}^{1/C}
  \] for some $C > 0$ depending on $\Phi$. This recovers and strengthens a
result of Gowers and Wolf in these cases. Moreover, the proof uses only
multiple applications of the Cauchy--Schwarz inequality, avoiding appeals to
the inverse theory of the Gowers norms.
  We also show that some dependence of $C$ on $\Phi$ is necessary; that is, the
constant $C$ can unavoidably become large as the coefficients of $\Phi$ grow.
