  Recent fluctuation identities for $\alpha$-stable L\'evy processes have
decomposed paths using generalised spherical polar coordinates revealing an
underlying Markov Additive Process (MAP) for which a more advanced form of
excursion theory can be exploited. Inspired by this approach, we give a
different decomposition of the $d$-dimensional isotropic $\alpha$-stable L\'evy
processes in terms of orthogonal coordinates. Accordingly we are able to
develop a number of $n$-tuple laws for first entrance into a half-space. We
also numerically construct the law of first entry of the process into a slab of
the form $(-1, 1)\times \mathbb{R}^{d-1}$ using a walk-on-half-spaces Monte
Carlo approach.
