Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of End F M, where M is a Yetter–Drinfeld module over B with dimB < ∞. In particular, generalized classical braided m-Lie algebras sl q, f (GM G (A), F) and osp q, t (GM G (A), M, F) of generalized matrix algebra GM G (A) are constructed and their connection with special generalized matrix Lie superalgebra sl s, f (GM Z_2(A s ), F) and orthosymplectic generalized matrix Lie super algebra osp s, t (GM Z_2(A s ), M s , F) are established. The relationship between representations of braided m-Lie algebras and their associated algebras are established.