In Schwartz' and Peskin's QFT books, when trying to deal with representations of the Lorentz group the authors study the representations of the Lie algebra of such group.
By definition, if $SO(1,3)$ is the Lorentz group, the Lie algebra is $\mathfrak{so}(1,3)$ defined as the set of all left-invariant vector fields on $SO(1,3)$ which in turns is equivalent to the tangent space at the identity of $SO(1,3)$.
Now, $SO(1,3)$ is a real manifold. Hence its tangent space at the origin is a real vector space.
Anyway, the books say that there are elements on this Lie algebra called generators defined by some complex matrices $J_i$ and $K_i$ such that any group element is
$$\Lambda = \exp(i\theta^i J_i+i\beta^i K_i)$$
and such that
$$[J_i,J_j]=i\epsilon_{ijk}J_k$$
$$[K_i,K_j]=-i\epsilon_{ijk}J_k$$
$$[J_i,K_j]=i\epsilon_{ijk}K_k$$
Now there's something quite wrong here. There are two main points I've noticed:
How can the elements of $\mathfrak{so}(1,3)$ be complex matrices if this is a real vector space? The matrices certainly need to be real. Unless one is hidden a complexification somewhere, but this isn't made clear in the books. If that's the case, where and why does one use a complexification?
It is not true that all elements of the group can be recovered by exponentiation, if I'm not mistaken. I really don't remember this quite well, but claiming that all elements have that form seems wrong. Furthermore, the exponentiation I know about is the map $\exp : \mathfrak{so}(1,3)\to SO(1,3)$ defined by
$$\exp(A)=\phi^{X^A}_1(e),$$
where $X^A$ is the associated left-invariant vector field, $\phi^{X^A}_t$ is its flow and $e\in SO(1,3)$ is the identity. If I'm not mistaken this exponential map isn't surjective. How the author is justified to say that any group element is of this form?
In summary how to connect the physicst's approach the author presents to the usual Lie group / Lie algebra theory?
