The Vertex Algebra M(1)+ and Certain Affine Vertex Algebras of Level 1

We give a coset realization of the vertex operatoralgebra $M(1)^+$ with central charge $ell$. We realize $M(1) ^+$as a commutant of certain affine vertex algebras of level $-1$ inthe vertex algebra $L_{C_{ell} ^{(1)}}(- frac{1}{2}Lambda_0)otimes L_{C_{ell} ^{(1)}}(- frac{1}{2}Lambda_0)$. We show thatthe simple vertex algebra $L_{C_{ell} ^{(1)}}(-Lambda_0)$ can be(conformally) embedded into $L_{A_{2 ell -1} ^{(1)}} (-Lambda_0)$and find the corresponding decomposition. We also study certaincoset subalgebras inside $L_{C_{ell} ^{(1)}}(-Lambda_0)$.