LAUSR

We give an explicit description for a weight three generator of the coset vertex operator algebra $${C_{{L_{\widehat {s\ln }}}\left( {l,0} \right) \otimes {L_{\widehat {s\ln }}}\left( {1,0} \right)}}\left( {{L_{\widehat {s\ln }}}\left( {l + 1,0} \right)} \right)$$CLsln^(l,0)⊗Lsln^(1,0)(Lsln^(l+1,0)), for n ≥ 2, l > 1. Furthermore, we prove that the commutant $${C_{{L_{\widehat {s\operatorname{l} 3}}}\left( {l,0} \right) \otimes {L_{\widehat {s\operatorname{l} 3}}}\left( {1,0} \right)}}\left( {{L_{\widehat {s\operatorname{l} 3}}}\left( {l + 1,0} \right)} \right)$$CLsl3^(l,0)⊗Lsl3^(1,0)(Lsl3^(l+1,0)) is isomorphic to the W-algebra $${W_{ - 3 + \frac{{l + 3}}{{l + 4}}}}\left( {sl3} \right)$$W−3+l+3l+4(sl3), which confirms the conjecture for the sl3 case that $${C_{{L_{\widehat g}}\left( {l,0} \right) \otimes {L_{\widehat g}}\left( {1,0} \right)}}\left( {{L_{\widehat g}}\left( {l + 1,0} \right)} \right)$$CLg^(l,0)⊗Lg^(1,0)(Lg^(l+1,0)) is isomorphic to $${W_{ - h + \frac{{l + h}}{{l + h + 1}}}}\left( g \right)$$W−h+l+hl+h+1(g) for simply-laced Lie algebras g with its Coxeter number h for a positive integer l.