I am confused about what's going on in the attached picture (from the introduction of Friedrich's Dirac Operators in Geometry). The author claims that for an operator $P$ to satisfy $P = \sqrt{\triangle}$ where $\triangle$ is the n dimensional Laplacian operator (with a minus sign) and $P = \Sigma_{i=1}^n \gamma_i \frac{\partial}{\partial x_i}$, it must satisfy $\gamma_i^2 = -E$ for $i$ from 1 to $n$ and $\gamma_i \gamma_j + \gamma_i \gamma_j = 0$ for $i \neq j$.
I can understand the latter condition as it makes the non-quadratic terms dissapear when we square $P$. But the former condition doesn't make any sense to me: E is defined earlier as the energy operator $ih\frac{\partial}{\partial t}$ which doesn't make any sense for recovering the Laplacian. It seems to me that the first condition should be $\gamma_i^2 = -1$ for all $i$ to get the Laplacian back. Can someone clarify where the $-E$ comes from?
