  A system of linear equations $L$ over $\mathbb{F}_q$ is common if the number
of monochromatic solutions to $L$ in any two-colouring of $\mathbb{F}_q^n$ is
asymptotically at least the expected number of monochromatic solutions in a
random two-colouring of $\mathbb{F}_q^n$. Motivated by existing results for
specific systems (such as Schur triples and arithmetic progressions), as well
as extensive research on common and Sidorenko graphs, the systematic study of
common systems of linear equations was recently initiated by Saad and Wolf.
  Building upon earlier work of Cameron, Cilleruelo and Serra, as well as Saad
and Wolf, common linear equations have recently been fully characterised by
Fox, Pham and Zhao, who asked about common \emph{systems} of equations. In this
paper we move towards a classification of common systems of two or more linear
equations. In particular we prove that any system containing an arithmetic
progression of length four is uncommon, confirming a conjecture of Saad and
Wolf. This follows from a more general result which allows us to deduce the
uncommonness of a general system from certain properties of one- or
two-equation subsystems.
