Proposition 4

For an ordered vector space $E$ to be the ordered direct sum of two supplementary linear subspaces $V, W$, it is necessary and sufficient that the relations $x \in V, y \in W, x + y \geq 0$ imply $x \geq 0$ and $y \geq 0$.

Since $x \geq 0$ and $y \geq 0$ imply $x + y \geq 0$ in $E$, the condition in the statement says that $(x, y) \mapsto x + y$ transforms the set of elements $\geq 0$ of $V \times W$ into the set of elements $\geq 0$ of $E$.