A characterization of quadratic forms of type \(E_6\), \(E_7\), and \(E_8\)
Quadratic forms are considered over fields of arbitrary characteristic. A quadratic form \(q\) is said to have a norm splitting if it has an orthogonal decomposition \[ q \simeq s_1N \perp s_2N \perp \cdots \perp s_dN \] with constants \(s_i\) in the base field and \(N\) the norm form from a separable quadratic extension of the base field. A quadratic form of dimension 6 (8, resp. 12) is said to be of type \(E_6\) (\(E_7\), resp. \(E_8\)) if it has a norm splitting (with the product of the constants not representable by the norm form, resp. with the negative product of constants represented by the norm form). The main result gives a characterization of the forms of each type in terms of its discriminant and Clifford invariant. Thus a 6-dimensional form is of type \(E_6\) if and only if it has a nontrivial discriminant and its Clifford invariant is determined by a quaternion algebra, which, if it is a division algebra, contains the discriminant extension of the form. Another equivalent condition is that the even Clifford algebra is isomorphic to the \(4 \times 4\) matrix algebra over a quadratic extension of the base field. Similar characterizations are provided for the types \(E_7\) and \(E_8\).
scientific article; zbMATH DE number 1836714