Let $\mathbb{C}l(n)$ denote the Clifford algebra over $\mathbb{C}^n$ with the standard bilinear form. Then $$\mathbb{C}l(n) \cong \begin{cases} \text{End}(\mathbb{C}^N) \quad n \text{ is even}\\ \text{End}(\mathbb{C}^N) \oplus \text{End}(\mathbb{C}^N) \quad n \text{ is odd}\end{cases} \tag{1}$$ where the value of $N$ depends on both the value of $n$ and if $n$ is even or odd. In Hamilton's Mathematical Gauge Theory he defines the vector space of Dirac spinors as $\mathbb{C}^N$ and the Dirac spinor representation of the complex Clifford algebra $$\rho: \mathbb{C}l(n) \rightarrow \text{End}(\mathbb{C}^N) \tag{2}$$ by (1) if $n$ is even and by $$\mathbb{C}l(n) \xrightarrow{\cong}\text{End}(\mathbb{C}^N) \oplus \text{End}(\mathbb{C}^N) \xrightarrow{\pi_1} \text{End}(\mathbb{C}^N) \tag{3}$$ if $n$ is odd. Here $\pi_1$ is the projection onto the first factor. This defines Dirac spinors.
Why do we project onto the first factor in (3) if $n$ is odd? Wouldn't the direct product in (1) be enough to define a representation (2)?