I'm reading Lounesto's CLifford Algebras and Spinors and on page 26 (also below) he states the following: \begin{align} C\mathcal{l}_2=\mathbb{R}\oplus\mathbb{R}^2\oplus\bigwedge^2\mathbb{R}^2. \end{align} I see that he states in the prior text a clue, i.e. "the Clifford algebra $C\mathcal{l}_2$ contains copies of $\mathbb{R}$ and $\mathbb{R}^2$, and it is the direct sum of its subspaces of elements of degrees $0,1,2$..."
How can I think of this (more simply) geometrically, i.e. this "direct sum" of the subspaces?
The page from the text:
You can think of $\mathcal{Cl}_2$ as a 4-dimensional vector space, with basis elements $1$, $\mathbf e_1$, $\mathbf e_2$, $\mathbf e_{12}$. The set of all vectors in $\mathbb R^2$ can then be seen as a two-dimensional subspace of the vector space $\mathcal{Cl}_2$, and the scalars and pseudoscalars each form 1-dimensional subspaces.