  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 discuss some basic properties and explain
structural differences to the associative Clifford analysis setting. The
non-associativity requires special attention in the definition of an
appropriate inner product and hence in the definition of a generalized Szeg\"o
projection. Whenever we want to apply classical theorems from reproducing
kernel Hilbert spaces we first need to switch to the consideration of
real-valued inner products where the Riesz representation theorem holds. Then
we introduce a generalization of the dual Cauchy transform for octonionic
monogenic functions which represents the adjoint transform with respect to the
real-valued inner product $\langle \cdot, \cdot \rangle_0$ together with an
associated octonionic Kerzman-Stein operator and related kernel functions.
  Also in the octonionic setting, the Kerzman-Stein operator that we introduce
turns out to be a compact operator. A motivation behind this approach is to
find an approximative method to compute the Szeg\"o projection of octonionic
monogenic functions offering a possibility 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.
