Factorizations of Eisenstein Series of Level up to 4

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 $4$.