  An algebraic formulation is given for the embedded noncommutative spaces over
the Moyal algebra developed in a geometric framework in \cite{CTZZ}. We
explicitly construct the projective modules corresponding to the tangent
bundles of the embedded noncommutative spaces, and recover from this algebraic
formulation the metric, Levi-Civita connection and related curvatures, which
were introduced geometrically in \cite{CTZZ}. Transformation rules for
connections and curvatures under general coordinate changes are given. A bar
involution on the Moyal algebra is discovered, and its consequences on the
noncommutative differential geometry are described.
