  Let $m$ be an odd positive integer and $D_m(\mathcal {A})$ be the
$m$-periodic derived category of a finitary hereditary abelian category
$\mathcal {A}$. In this note, we prove that there is an embedding of algebras
from the derived Hall algebra of $D_m(\mathcal {A})$ defined by Xu-Chen [13] to
the extended derived Hall algebra of $D_m(\mathcal {A})$ defined in [16]. This
homomorphism is given on basis elements, rather than just on generating
elements.
