  The aim of this paper is to study some brackets defined on
$(\tau,\sigma)$-derivations satisfying quasi-Lie identities. Moreover, we
provide examples of $(p,q)$-deformations of Witt and Virasoro algebras as well
as $\mathfrak{sl}(2)$ algebra. These constructions generalize the results
obtained by Hartwig, Larsson and Silvestrov on $\sigma$-derivations, arising in
connection with discretizations and deformations of algebras of vector fields.
