  We show that the most standard Eisenstein series such as $E_4(\tau)$ or
$2E_2(2\tau)-E_2(\tau)$, and also the function $\theta^2(\tau)$, are in a
natural way the product of two conjugate Eisenstein series of half their weight
and double their level, as well as a number of similar elementary identities
for $E_6$ and Eisenstein series of levels $2$, $3$, and $4$.
