An algebra in which any two elements generate a Lie subalgebra is called binary-Lie. Over a field of characteristic \(\neq 2\), such algebras are defined by the identities \(x^ 2=0\) and \(((xy)y)x+((yx)x)y=0\). An algebra is called algebraic if every right multiplication operator satisfies a polynomial equation; and if the degrees of these equations have an overall bound, then the algebra is said to be algebraic of bounded index. Also, an algebra is called weakly algebraic if the orbits of the right multiplication operators are finite-dimensional. Assuming all algebras to be over an algebraically closed field of characteristic 0, the main results of this work are as follows: (1) A weakly algebraic binary-Lie algebra which satisfies the maximal condition on abelian subalgebras is finite-dimensional. (2) A binary-Lie algebra is algebraic of bounded index if and only if it contains an Engel ideal of finite codimension which is algebraic of bounded index. (3) Let \(Q\) be a binary-Lie algebra which is algebraic of bounded index. Then there is an almost semisimple subalgebra \(P\) and a nilpotent ideal \(N\) such that \(Q=P+N+G\), where \(G\) is a subalgebra such that \(G^ 2\) lies in an Engel ideal of \(G\) of finite codimension, and \(PG=0\). (Also, when \(Q\) is actually a Lie algebra, then it has a stronger characterization).