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Using the Zhelobenko's approach we investigate a branching of an irreduciblerepresentation of $g_n$ under the restriction of algebras $g_n\downarrowg_{n-1}$, where $g_n$ is a Lie algebra of type $B_n$, $C_n$, $D_n$ or a Liealgebra of type $A$, where in this case we put $g_{n}=\mathfrak{gl}_{n+1}$,$g_{n-1}=\mathfrak{gl}_{n-1}$. We give a new explicit description of the spaceof the $g_{n-1}$-highest vectors, then we construct a base in this space. Thecase $n=2$ is considered separately for different algebras, but a passage from$n=2$ to an arbitrary $n$ is the same for all series $A$, $B$, $C$, $D$. Thisnew procedure has the following advantage: it establishes a relation betweenspaces of $g_{n-1}$-highest vectors for different series of algebras. Thisprocedure describes an extension of Gelfand-Tsetlin tableaux to the left.

A unified approach to construction of Gelfand-Tsetlin-Zhelobenko base vectors for series $A$, $B$, $C$, $D$