On primitive solutions of quadratic Diophantine equation with four variables

We treat certain integral lattices in ternary quadratic spaces. Each of them is described by an order associated with the lattice in the even Clifford algebra. In this viewpoint we determine the mass of the genus of such a lattice. As an application the number of primitive solutions of a quadratic Diophantine equation with four variables can be derived by means of the mass formula due to Shimura.