Consider a basis $(c_1 ^ {\dagger}, c_2 ^ {\dagger}, c_1 ^ {\dagger}, c_1, c_2, c_3 )$ of creation and annihilation operators for $W=V \oplus V^*$.
I need help to write the basis for Clifford algebra $Cl^2 (V \oplus V^*)$=$Cl^2 (W)$. By definition, $$ Cl^2 (V \oplus V^*) := \bigoplus_{l =0 }^2 T^l(W)/I, $$ where $I$ is two sided ideal generated by $(xy+yx)-b(x,y).1$ where $b$ is the canonical bilinear form attached to $W.$
I wrote $$ Cl^2 (V \oplus V^*) = \mathbb{C}/I \oplus W / I \oplus W \otimes W/I. $$
Is this correct?
What is the basis for $Cl^2 (V \oplus V^*)$? Also, how can I write basis for quotient space $Cl^3(W)/Cl^2(W)$? For the latter, I am assuming that it is enough to understand the basis for $Cl^3(W)$. Then I can simply consider the coset of each basis element. Right?
I'm not familiar with the context of your question, but the Clifford algebra of a direct sum can easily be written using the graded tensor product. If $W=V_1\oplus V_2$ with bilinear forms $b,b_1$ and $b_2$ respectively ($b(x,y) = b_1(x_1,y_1) + b_2(x_2,y_2)$). Then, $$ \operatorname{Cl}(W,b) \simeq \operatorname{Cl}(V_1,b_1)\hat{\otimes}\operatorname{Cl}(V_2,b_2). $$ With this I believe that: $$ \operatorname{Cl}^i(W,b) \simeq \bigoplus_{j+k=i}\operatorname{Cl}^j(V_1,b_1)\hat{\otimes}\operatorname{Cl}^k(V_2,b_2). $$