  We show that if $X\subseteq \mathbb{P}^{n-1}$, defined over $\mathbb{Q}$ by a
cubic form that splits off two forms, with $n\geq 11$, then $X(\mathbb{Q})$ is
non-empty. The same holds for an $(m_1,m_2)$-form with $m_1\geq 4$ and $m_2\geq
5$.
