Consider the commutative, unital algebras $\mathbb{R}(i), \mathbb{R}(\epsilon)$ and $\mathbb{R}(\eta)$, where the adjunctions satisfy $i^{2} = -1, \epsilon^{2} = 0$ and $\eta^{2} = 1$ (but $i, \epsilon$ and $\eta$ are not elements of $\mathbb{R}$). Since the operations of addition and multiplication are continuous in the corresponding product topologies, these algebras are examples of topological rings of hypercomplex numbers.
It is clear that $\mathbb{R}(i) \cong \mathbb{C}$ and, with a little work, one can prove $\mathbb{R}(\epsilon) \cong \bigwedge \mathbb{R}$, the exterior algebra of the vector space $\mathbb{R}$ (over the field $\mathbb{R}$) and also $\mathbb{R}(\eta) \cong \mathbb{R} \oplus \mathbb{R}$, where the explicit bijection is a lift of the map $a + b \eta \mapsto (a+b, a- b)$. The latter two are not fields because they contain non-trivial nilpotent elements, e.g., $b\epsilon $ and $\frac{1}{2}(1-\eta)$.
Since there is clearly some relationship between the algebras, does $\mathbb{R}(\eta)$ admit an interpretation in terms of $S(\mathbb{R})$, the symmetric algebra of the vector space $\mathbb{R}$ over the field $\mathbb{R}$ or some similar structure?