Lemmas to characterize Eistenstein primes Here are lemmas to characterize Gaussian primes

Lemma1 Let $p$ be an odd prime such that $p\equiv 1 \pmod 4$. Then, the Legendre symbol $(-1/p)=1$
Lemma2 Let $p$ be an odd prime such that $p\equiv 1\pmod 4$. Then, there are integers $a,b$ such that $p=a^2+b^2$.

Lemma1 implies that $p$ is not a prime in the Gaussian integers and hence $p$ can be written as a product of two primes $\pi,\pi'$. Then, this implies that the norm of $\pi$ is $p$, hence proving the Lemma 2. Then, Lemma2 characterized Gaussian primes easily.
Just like these lemmas for characterizing Gaussian primes, I want to know lemmas for characterizing Eisenstein primes before I prove it by myself. (I don't want to waste my time too much)
So, what are meaningful lemmas to prove the below statement?

Let $\pi$ be an Eisenstein prime such that $|\pi|^2$ is not a prime in $\mathbb{Z}$. Then $\pi$ associates to a prime $p$ in $\mathbb{Z}$ such that $p\equiv 2 \pmod 3$. Also the converse holds.

Thank you in advance :)

In the book of Ireland and Rosen the primes in the ring of Eisenstein integers $\mathbb{Z}[\omega]$ are classified as follows:
Proposition: Suppose that $p$ and $q$ are primes in $\mathbb{Z}$. If $q\equiv 2(3)$, then $q$ is prime in $\mathbb{Z}[\omega]$. If $p\equiv 1(3)$, then $p=N(\pi)$, where $\pi$ is a prime in $\mathbb{Z}[\omega]$. (Finally, $3=-\omega^2(1-\omega)^2$, and $1-\omega$ is prime.)
The proof uses quadratic reciprocity for $p\equiv 1(3)$ (to show that $(-3/p)=1$ for such primes), and uses the norm in $\mathbb{Z}[\omega]$ (i.e., $p=x^2-xy+y^2=N(\pi)$, $\pi=x+y\omega$).
