So far I met *three* definitions of the so called *generalized Dirac operator*(or *Dirac type* operators. Everything takes place over Riemannian manifols $M$ and we have smooth hermitian vector bundle $S \to M$ over $M$.

**First definition** The first order differential operator $D$ is called Dirac type if its symbol $\sigma$ has the property:
$$\sigma(x,\xi)^2u=-\|\xi\|^2u$$ for $x \in M, \xi \in T^*_xM, u \in S_x$.

**Second definition** $D$ is called Dirac type operator if $D^2$ is of the form $$\sum_{i,j}g^{ij}\partial_i\partial_j$$ modulo the lower order terms (here $g^{ij}$ are components of Riemannian metric on cotangent bundle).

**Third definition** $D$ is called Dirac type if there is a Clifford action $c$ and a Clifford connection $\nabla$ (i.e. metric connection compatible with this Clifford action) such that $D=c \circ \nabla$ (some authors put $i$ in front of this operator).

Are these definitions equivalent? If so, why is it true?

The one implication which is evident for me is that the second definition implies the first