Flat Hypercomplex Nilmanifolds are \(\mathbb H\)-Solvable

Let \(\mathbb H\) be a quaternion algebra generated by \(I,J\) and \(K\) . We say that a hypercomplex nilpotent Lie algebra \(\mathfrak g\) is \(\mathbb H\) -solvable if there exists a sequence of \(\mathbb H\) -invariant subalgebras containing \(\mathfrak g_{i+1}=[\mathfrak g_i,\mathfrak g_i]\) , \(\mathfrak g=\mathfrak g_0\supset\mathfrak g_1^{\mathbb H}\supset\mathfrak g_2^{\mathbb H}\supset\cdots\supset\mathfrak g_{k-1}^{\mathbb H}\supset\mathfrak g_k^{\mathbb H}=0,\) such that \([\mathfrak g_i^{\mathbb H},\mathfrak g_i^{\mathbb H}]\subset\mathfrak g^{\mathbb H}_{i+1}\) and \(\mathfrak g_{i+1}^{\mathbb H}=\mathbb H[\mathfrak g_i^{\mathbb H},\mathfrak g_i^{\mathbb H}] \) . Let \(N=\Gamma\setminus G\) be a hypercomplex nilmanifold with the flat Obata connection and \(\mathfrak g=\operatorname{Lie}(G)\) . We prove that the Lie algebra \(\mathfrak g\) is \(\mathbb H\) -solvable.