Expressing complexified tangent bundle of a spin 4-manifold as a Hom bundle

Question

I am reading Moore's book Lectures on Seiberg-Witten Invariants, section 2.2. First here are some defintions that the book uses.

1. The group $\operatorname{Spin}(4)$ is defined to be the product group $SU(2)\times SU(2)$.

2. Let $V$ be the 4-dimensional $\Bbb R$-algebra with $\Bbb R$-basis $\textbf{1}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \textbf{i}=\begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}, \textbf{j}=\begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}, \textbf{k}=\begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} $. Thus $V$ consists of $2\times 2$ complex matrices of the form $\begin{bmatrix} a & -\bar{b} \\ b & \bar{a} \end{bmatrix}$.

3. Define $\rho:\text{Spin}(4)\to GL(V)$ by $\rho(A,B)(Q)=BQA^{-1}$. If we identify $V$ and $\Bbb R^4$ (as $\Bbb R$-vector spaces) by $\textbf{1}\leftrightarrow (1,0,0,0), \textbf{k}\leftrightarrow (0,1,0,0), \textbf{i}\leftrightarrow (0,0,1,0), \textbf{j}\leftrightarrow (0,0,0,1) $, it can be seen that $\rho$ maps $\text{Spin}(4)$ into $SO(4)$. Also let $\rho_+, \rho_-:\text{Spin}(4)\to SU(2)$ be the projections onto first and second components, respectively.

4. Lastly, here is the definition of a spin structure on a 4-manifold. Suppose $M$ is an oriented Riemannian manifold of dimension $4$. Then we can choose trivializing open cover $\{U_\alpha\}$ for $TM$ such that the corresponding transition functions $g_{\alpha\beta}$ take values in $SO(4)$. A spin structure on $M$ is collection of maps $\tilde{g}_{\alpha\beta}:U_\alpha \cap U_\beta \to \text{Spin}(4)$ such that $\rho \circ \tilde{g}_{\alpha\beta}=g_{\alpha\beta}$ and $\tilde{g}_{\alpha\beta}\tilde{g}_{\beta\gamma}=\tilde{g}_{\alpha\gamma}$.

Now here is my question. Suppose we are given a spin structure on $M$. Considering $\{\rho_+\circ \tilde{g}_{\alpha\beta}:U_\alpha\cap U_\beta \to SU(2)\}$ and $\{\rho_-\circ \tilde{g}_{\alpha\beta}:U_\alpha\cap U_\beta \to SU(2)\}$, we get two $\Bbb C^2$-bundles $W_+$ and $W_-$ over $M$. The book is claiming that the complexified tangent bundle $TM\otimes \Bbb C$ is isomorphic to the bundle $E:=\text{Hom}_{\Bbb C}(W_+,W_-)$, but I can't see why. Why are these two bundles are isomorphic?

I think comparing transition functions is an appropriate approach. First, the bundle $TM\otimes \Bbb C$ has transition functions $g_{\alpha\beta}$. If we write $\tilde{g}_{\alpha\beta}=(A_{\alpha\beta},B_{\alpha\beta})$, then $W_+$ and $W_-$ have transition functions $A_{\alpha\beta}$ and $B_{\alpha\beta}$, respectively, so $E$ has transition functions $((A_{\alpha\beta})^{-1})^T \otimes B_{\alpha\beta}$. Since $A_{\alpha\beta}$ take values in $SU(2)$, $((A_{\alpha\beta})^{-1})^T=(A_{\alpha\beta}^*)^T=\overline{A_{\alpha\beta}}$. Thus $E$ has transition functions $\overline{A_{\alpha\beta}}\otimes B_{\alpha\beta}$. How can we show that this is equivalent to $g_{\alpha\beta}$?