  The aim of this paper is to extend the so called slice analysis to a general
case in which the codomain is a real vector space of even dimension, i.e. is of
the form $\mathbb{R}^{2n}$. We define a cone $\mathcal{W}_\mathcal{C}^d$ in
$[End(\mathbb{R}^{2n})]^d$ and we extend the slice-topology $\tau_s$ to this
cone. Slice regular functions can be defined on open sets in
$\left(\tau_s,\mathcal{W}_\mathcal{C}^d\right)$ and a number of results can be
proved in this framework, among which a representation formula. This theory can
be applied to some real algebras, called left slice complex structure algebras.
These algebras include quaternions, octonions, Clifford algebras and real
alternative $*$-algebras but also left-alternative algebras and sedenions, thus
providing brand new settings in slice analysis.
