Some groups of type $E_{7}$

Abstract

An algebraic group of type $E_{7}$ over an algebraically closed field has an irreducible representation in a vector space of dimension $56$ and is, in fact, the identity component of the automorphism group of a quartic form on the space. This paper describes the construction of the quartic form if the characteristic is $\neq 2, 3$, taking into account a field of definition $F$. Certain $F$-forms of $E_{7}$ appear in the automorphism groups of quartic forms over $F$, as well as forms of $E_{6}$. Many of the results of the paper are known, but are perhaps not easily accessible in the literature.