We say that a G-structure induces a minimal left ideal for a Clifford algebra of the form \({{\mathbb {R}}}_{p,q}\) if, by means of the vector space isomorphism between the space of exterior forms \(\bigwedge ^{*}({{\mathbb {R}}}^{p+q})^{*}\) and the Clifford algebra \({{\mathbb {R}}}_{p,q}\) , we can express the primitive idempotent f that defines the minimal left ideal \({{\mathbb {R}}}_{p,q} f\) as the image of an algebraic expression of the model tensor that defines the special geometric structure in the space of exterior forms. In this paper we focus on special geometric structures in dimensions 6, 7, and 8 and the minimal left ideals that they induce for the Clifford algebras \({{\mathbb {R}}}_{0,6},{{\mathbb {R}}}_{0,7},\) and \({{\mathbb {R}}}_{0,8}\) .
