# Lipschitz primes

A Lipschitz integer is a Quaternion with integer coefficients. The norm is defined as $$N(a+ib+jc+kd)=a^2+b^2+c^2+d^2$$ which is a multiplicative function

$$N:\mathbb H\to\mathbb R$$, $$N(\alpha\beta)=N(\alpha)N(\beta)$$

Now since any natural number can be written as the sum of four squares of integers, all prime numbers in $$\mathbb N$$ are composite Lipschitz integers

$$a^2+b^2+c^2+d^2=(a+ib+jc+kd)(a-ib-jc-kd)$$

Obviously, if $$\alpha$$ is a composite, then so is $$N(\alpha)$$.

In the ring of Gaussian integers it also holds that if both the real and the imaginary terms are non-zero, then the norm is prime if the Gaussian number is prime. My question is

If a Lipschitz integer is prime and all four terms are non-zero, is then the norm a prime ?