Composition laws of binary quadratic forms and isolations of quadratic forms

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.