# Graded tensor product in Clifford algebras

I'm trying to understand the proof of Proposition 1.5 in Spin Geometry by H. B. Lawson, JR. and M.-L. Michelsohn.

This says that if $$V = V_1 \oplus V_2$$ is an $$q$$-orthogonal decomposition of the vector space $$V$$ where $$(V_i, q_i)$$ are quadratic spaces with quadratic forms $$q_i$$ for $$i=1,2$$ and $$q=q_1 \oplus q_2$$. Then, there is a natural isomorphism of Clifford algebras $$Cl(V,q) \simeq Cl(V_1,q_1) \phantom{.} \hat{\otimes} \phantom{.} Cl(V_2,q_2).$$ where $$\hat{\otimes}$$ denotes the $$\mathbb{Z}_2$$-graded tensor of algebras.

They consider the map $$f\colon V_1\bigoplus V_2 \rightarrow Cl(V_1,q_1) \phantom{.} \hat{\otimes} \phantom{.} Cl(V_2,q_2)$$, such that $$v_1+v_2$$ is sent to $$v_1\otimes 1 + 1\otimes v_2$$, where $$e_i \in V_i$$ for $$i=1,2$$. Now $$f(v_1+v_2)^2=(v_1\otimes 1 + 1\otimes v_2)^2=v_1^2\otimes1 + 1\otimes v_2^2 + v_1\otimes v_2 \\ + (-1)^{\mathrm{deg}(v_1)\mathrm{deg}(v_2)} v_1\otimes v_2= -(q_1(v_1)+q_2(v_2)) 1\otimes 1.$$

I can not understand this last inequality since I don't see how can $$v_1\otimes v_2 + (-1)^{\mathrm{deg}(v_1)\mathrm{deg}(v_2)} v_1\otimes v_2$$ cancel.

Elements of $$V_1$$ and $$V_2$$ always have degree $$1$$ in the Clifford algebra, so $$\text{deg}(v_1) \text{deg}(v_2) = 1$$.