Contact manifold

A $(2n + 1)$-dimensional manifold $M$ is said to be a almost contact manifold if it admits a global 1-form $\eta$ such that $\eta\wedge (d\eta)^n\neq 0$.

There is an equivalent definition by J. W. Gray:

A manifold is almost contact structure if the structural group of the tangent bundle is reducible to $U(n)\times 1$.

Contact pairs manifold

A pair $(\alpha_1, \alpha_2)$ of $1$-forms on a manifold is said to be a contact pair of type $(h,k)$ if $$\alpha_1 \wedge (d\alpha_1)^h \wedge\alpha_2 \wedge(d\alpha_2)^k\ \text{ is a volume form,}\\ (d\alpha_1)^{h+1} = 0\ \text{and}\ (d\alpha_2)^{k+1} = 0.$$

Question: Is there an equivalent definition for contact pair similar to contact case?

Thanks.