<p>A monogenic function of two vector variables is a function annihilated by two Dirac operators. We give the explicit form of differential operators in the Dirac complex resolving two Dirac operators and prove its ellipticity directly. This opens the door to apply the method of several complex variables to investigate this kind of monogenic functions. We prove the Poincaré lemma for this complex, i.e. the non-homogeneous equations are solvable under the compatibility condition, by solving the associated Hodge Laplacian equations of fourth order. As corollaries, we establish the Bochner–Martinelli integral representation formula for two Dirac operators and the Hartogs’ extension phenomenon for monogenic functions. We also apply abstract duality theorem to the Dirac complex to obtain the generalization of Malgrange’s vanishing theorem and establish the Hartogs–Bochner extension phenomenon for monogenic functions under the moment condition.</p>