  The aim of this paper is to find generating sets of commuting involutions and
use them to explicitly construct minimal representations of Clifford algebras
$Cl(n)_{p,q}$. By results of [HL] and [LW], we know the dimension of such
minimal representations, which is linked to the maximal number of commuting
involutions in the algebra, dependent only on $p$ and $q$. We provide an
algorithm to construct these generating sets of involutions explicitly for all
Clifford algebras $Cl(n)_{p,q}$ and provide some examples. Involutions yield
mutually non-annihilating idempotents whose product gives a projection map
$P^+$ with image $Cl(n)P^+$ being a minimal left ideal, which is a spinor
space. Using the projections, we find minimal representations of Clifford
algebras by combining matrices of left multiplication endomorphisms in the
spinor spaces. Finally, we provide examples showing calculations of minimal
representations.
