In this paper, first we introduce a new notion of commuting condition that $\phi {\phi }_{1}A=A{\phi }_1\phi$ between the shape operator $A$ and the structure tensors $\phi$ and ${\phi }_1$ for real hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$. Suprisingly, real hypersurfaces of type $\left(A\right)$, that is, a tube over a totally geodesic $G_2\left({ℂ}^{m+1}\right)$ in complex two plane Grassmannians $G_2\left({ℂ}^{m+2}\right)$ satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$ satisfying the commuting condition. Finally we get a characterization of Type $\left(A\right)$ in terms of such commuting...

A representation field for a non-maximal order $ℌ$ in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders containing a copy of $ℌ$. Not every non-maximal order has a representation field. In this work we prove that every commutative order has a representation field and give a formula for it. The main result is proved for central simple algebras over arbitrary global fields.

A commutative order in a quaternion algebra is called selective if it embeds into some, but not all, of the maximal orders in the algebra. It is known that a given quadratic order over a number field can be selective in at most one indefinite quaternion algebra. Here we prove that the order generated by a cubic root of unity is selective for any definite quaternion algebra over the rationals with type number 3 or larger. The proof extends to a few other closely related orders.