# Clifford Algebra Isomorphic to Exterior Algebra

Let $E$ be a vector space over a field $k$ and $Q$ be a quadratic form, that is, $$Q:E\to k$$ such that $$Q(\lambda e)=\lambda^2Q(e)\forall\lambda\in k\,e\in E$$

and such that $P_Q:E^2\to k$ is bilinear, where $$P_Q(e_1,e_2):=\frac{1}{2}\left[Q(e_1+e_2)-Q(e_1)-Q(e_2)\right]$$

We define the tensor algebra $T(E)$ from $E$ as $$T(E):=\bigoplus_{n=0}^{\infty}E^{\otimes n}$$where $E^{\otimes 0}\equiv k$ and $E^{\otimes n}\equiv E\otimes\dots\otimes E$ (n factors) with the product $$(e_1\otimes\dots\otimes e_n)\cdot(\tilde{e}_1\otimes\dots\otimes \tilde{e}_\tilde{n}) := e_1\otimes\dots\otimes e_n\otimes\tilde{e}_1\otimes\dots\otimes \tilde{e}_\tilde{n} \in E^{\otimes (n+\tilde{n})}$$extended to the whole of $T(E)$ by requiring linearity.

Let $I(E,Q)\subseteq T(E)$ be the ideal generated by the set $$\{ e\otimes e-Q(e)\cdot1_k \,|\, e\in E\}$$

The the Clifford algebra associated with $E$ and $Q$ is defined as $$Cl(E,Q) := T(E)/I(E,Q)$$

In Atiyah et al's monograph "Clifford Modules" (page 5 point (1.4)) it is claimed that $$G(Cl(E,Q)) \cong\Lambda(E)$$where $$\Lambda(E)\equiv T(E)/J(E)$$ where $J(E)\subseteq T(E)$ is the ideal generated by the set $$\{ e\otimes e \,|\, e\in E\}$$ and $G$ is the associated graded algebra to the filtering of $Cl(E,Q)$ which is induced by the filtering of $T(E)$ given by $$F^q T(E):=\bigoplus_{n=0}^{q}E^{\otimes n}$$ for all $q\in\mathbb{N}_{\geq0}$

This isomorphism is only as vector spaces, not as algebras.

My question is: can anyone please write down explicitly what this isomorphism is and describe it? In Wikipedia they give another explicity isomorphism between $Cl(E,Q)$ itself and $\Lambda(E)$, mention the one I'm after, but do not provide it. Because of the complicated nature of the associated graded algebra to a filtered algebra, I'm a bit stumped by this.