  In this paper we consider generalized Hardy spaces in the octonionic setting
associated to arbitrary Lipschitz domains where the unit normal field exists
almost everywhere. First we analyze some basic properties and discuss
structural differences to the associative Clifford analysis setting. Then we
introduce a dual Cauchy transform for octonionic monogenic functions together
with an associated octonionic Kerzman-Stein operator and related kernel
functions. The non-associativity requires a special attention and sometimes
essentially different ideas to arrive at many fundamental statements; in
particular it requires a special form of the definition of the inner product.
Nevertheless, our adapted constructions are compatible with the classical
representations when associativity permits us to interchange the order of the
parenthesis.
  Also in the octonionic setting, the Kerzman-Stein operator that we introduce
turns out to be a compact operator and allows us to obtain approximations of
the Szeg\"o projection of octonionic monogenic functions. This in turn
represents a tool to tackle BVP in the octonions without the explicit knowledge
of the octonionic Szeg\"o kernel which is extremely difficult to determine in
general. We also discuss the particular cases of the octonionic unit ball and
the half-space. Finally, we relate our octonionic Kerzman-Stein operator to the
Hilbert transform and particularly to the Hilbert-Riesz transform in the
half-space case.
