  In this paper we study minimal algebraic surfaces with $p_g=q=1,K^2=4$ and
nonhyperelliptic Albanese fibrations of genus 4. We construct for the first
time a family of such surfaces as complete intersections of type $(2,3)$ in a
$\mathbb{P}^3$-bundle over an elliptic curve. For the surfaces we construct
here, the direct image of the canonical sheaf under the Albanese map is
decomposable (which is a topological invariant property).
  Moreover we prove that, all minimal surfaces with $p_g=q=1,K^2=4$ and
nonhyperelliptic Albanese fibrations of genus 4 such that the direct image of
the canonical sheaf under the Albanese map is decomposable are contained in our
family. As a consequence, we show that these surfaces constitute a
4-dimensional irreducible subset $\mathcal{M}$ of $\mathcal{M}_{1,1}^{4,4}$,
the Gieseker moduli space of minimal surfaces with $p_g=q=1, K^2=g=4$.
Moreover, the closure of $\mathcal{M}$ is an irreducible component of
$\mathcal{M}_{1,1}^{4,4}$.
