lie algebra - Why complexify in order to construct Dirac representation?

Suppose we have a theory is covariant under the Spin group Spin(2n-1; 1). We consider the real vector space $V = R^{2n-1,1}$, which naturally comes with a Lorentzian inner product. On this vector space we introduce an orthonormal basis $e_0; e_1; ... ; e_{2n-1}$, where $e _0$ denotes the time direction.

To construct the Dirac representation of Spin(2n-1; 1) we take the complexified space $T = \mathbb{C} $. My question is why is it that in order to construct the Dirac representation we complexify the space?

NOTE: Theory is even dimensional and of Lorentzian signature.