Quadro-quadric cremona transformations in low dimensions via the JC-correspondence

It has been previously established that a Cremona transformation of bidegree (2, 2) is linearly equivalent to the projectivization of the inverse map of a rank 3 Jordan algebra. We call this result the "JC-correspondence". In this article, we apply it to the study of quadro-quadric Cremona transformations in low-dimensional projective spaces. In particular we describe new very simple families of such birational maps and obtain complete and explicit classifications in dimension 4 and 5.