Isometries of semi-orthogonal forms on a $\mathbb Z$-module of rank 3

We study the isometry groups of semi-orthogonal forms (that is, forms whose Gram matrix in some basis is upper triangular with ones on the diagonal) on a $\mathbb Z$-module of rank 3. Such forms have a discrete parameter: the height (the trace of the dualizing operator + 3). We prove that the isometry group is either $\mathbb Z$ or $\mathbb Z_2\times\mathbb Z$, list all the cases when it is a direct product and describe the generator of order 2 in that case. We also describe a generator of infinite order for many particular values of the height.