  Let $\A$ be a finitary hereditary abelian category and $D(\A)$ be its reduced
Drinfeld double Hall algebra. By giving explicit formulas in $D(\A)$ for left
and right mutations, we show that the subalgebras of $D(\A)$ generated by
exceptional sequences are invariant under mutation equivalences. As an
application, we obtain that if $\A$ is the category of finite dimensional
modules over a finite dimensional hereditary algebra, or the category of
coherent sheaves on a weighted projective line, the double composition algebra
of $\A$ is generated by any complete exceptional sequence. Moreover, for the
Lie algebra case, we also have paralleled results.
