Composition laws of binary quadratic forms and isolations of quadratic forms

Abstract: A positive definite and integral quadratic form $f$ is called irrecoverable if there is a quadratic form $F$ such that it represents all proper subforms of $f$, whereas it does not represent $f$ itself. In this case, $F$ is called an isolation of $f$. In this article, we prove that there does not exist a binary isolation of any unary quadratic form. We also prove that there does not exist a ternary isolation of any binary quadratic form. Furthermore, if the form class group of a primitive binary quadratic form has no element of order $4$, then the discriminant of any quaternary isolation of it, if exists, is a square of an integer. The composition laws of primitive binary quadratic forms play an essential role in the proofs of the results.