Holzer reduction of solutions of quadratic ternary forms

Question

Suppose $(x_{0}, y_{0}, z_{0})$ is a solution to the equation $ax^2 + by^2 + cz^2 = 0$. The solution is said to be Holzer reduced if $x_{0} < \sqrt{|bc|}$, $y_{0} < \sqrt{|ac|}$ and $z_{0} < \sqrt{|ab|}$. It is a proven fact that given any solution $(x, y, z)$ to the above equation, we can construct a solution $(x_{0}, y_{0}, z_{0})$ which is Holzer reduced. Does anyone know of a better algorithm for this? I found one here in page 8 but it's a mere sketch. Thanks in advance.