  In 1997, Bousquet-M\'elou and Eriksson stated a broad generalization of
Euler's distinct-odd partition theorem, namely the $(k,l)$-Euler theorem. Their
identity involved the $(k,l)$-lecture-hall partitions, which, unlike usual
difference conditions of partitions in Rogers-Ramanujan type identities,
satisfy some ratio constraints. In a 2008 paper, in response to a question
suggested by Richard Stanley, Savage and Yee provided a simple bijection for
the $l$-lecture-hall partitions (the case $k=l$), whose specialization in $l=2$
corresponds to Sylvester's bijection. Subsequently, as an open question, a
generalization of their bijection was suggested for the case $k,l\geq 2$. In
the spirit of Savage and Yee's work, we provide and prove in this paper slight
variations of the suggested bijection, not only for the case $k,l\geq 2$ but
also for the cases $(k,1)$ and $(1,k)$ with $k\geq 4$. Furthermore, we show
that our bijections equal the recursive bijections given by Bousquet-M\'elou
and Eriksson in their recursive proof of the $(k,l)$-lecture hall and finally
provide the analogous recursive bijection for the $(k,l)$-Euler theorem.
