  We present methods for computing the explicit decomposition of the minimal
simple affine $W$-algebra $W_k(\mathfrak g, \theta)$ at a conformal level $k$
as a module for its maximal affine subalgebra $\mathcal V_k(\mathfrak
g^{\natural})$. A particular emphasis is given on the application of affine
fusion rules to the determination of branching rules. In almost all cases when
$\mathfrak g^{\natural}$ is a semisimple Lie algebra, we show that, for a
suitable conformal level $k$, $W_k(\mathfrak g, \theta)$ is isomorphic to an
extension of $\mathcal V_k(\mathfrak g^{\natural})$ by its simple module. We
are able to prove that in certain cases $W_k(\mathfrak g, \theta)$ is a simple
current extension of $\mathcal V_k(\mathfrak g^{\natural})$. In order to
analyze more complicated non simple current extensions at conformal levels, we
present an explicit realization of the simple $W$-algebra $W_{k}(sl(4),
\theta)$ at $k=-8/3$. We prove, as conjectured in arXiv:1407.1527, that
$W_{k}(sl(4), \theta)$ is isomorphic to the vertex algebra $\mathcal R^{(3)}$,
and construct infinitely many singular vectors using screening operators. We
also construct a new family of simple current modules for the vertex algebra
$V_k (sl(n))$ at certain admissible levels and for $V_k (sl(m | n)), m\ne n,
m,n\geq 1$ at arbitrary levels.
