Finding every representation of the Lorentz Group For some time I've been trying to truly understand how to find all the finite-dimensional representations of the proper orthochronous Lorentz Group $SO(1,3)$, the part connected to the identity. I have found no single reference that approach it from start to finish (the closest being the Wikipedia article) and although there are good books on Representation Theory which cover everything needed, I'm not sure I have grasped the whole thing. My current line of thought is the following (keep in mind that this is from a physicist point of view):
1) The group is non-compact, which means that there are no finite-dimensional unitary representations; and non-simply-connected, which means that there is no guarantee that there exists a ('global') representation of $SO(1,3)$ for each representation of $\mathfrak{so}(1,3)$ (we must look for its projective representations or regular representations of its universal covering).
2) We start complexifying the algebra: $\mathfrak{so}(1,3) \hookrightarrow \mathfrak{so}(1,3)_\mathbb{C}$. This is harmless since we can canonically extend every representation of an algebra to a unique complex-linear representation of its complexification; and, on the other way, we can restrict every representation of an algebra to a representation of the appropriate real form.
3) Then we may check a well-known change of basis which is enough to convince ourselves that $\mathfrak{so}(1,3)_\mathbb{C} \cong \mathfrak{su}(2)_\mathbb{C} \oplus \mathfrak{su}(2)_\mathbb{C}$ and, since the extension of the vector space of traceless hermitian matrices by the addition of traceless anti-hermitian matrices gives us the space of traceless complex matrices (or the other way around, in the math convention), we also have $\mathfrak{su}(2)_\mathbb{C} \oplus \mathfrak{su}(2)_\mathbb{C} \cong \mathfrak{sl}(2,\mathbb{C}) \oplus \mathfrak{sl}(2,\mathbb{C}) \cong \mathfrak{sl}(2,\mathbb{C})_\mathbb{C}$.
4) This is nice because  $\mathfrak{su}(2) \oplus \mathfrak{su}(2) \cong \mathfrak{sl}(2,\mathbb{C})$ is simply connected and we do know how to find its every irreducible represention by the highest weight construction.
5) Since $\mathfrak{sl}(2,\mathbb{C})$ is simply connected, we know that the isomorphism between $\mathfrak{so}(1,3)_\mathbb{C} \cong \mathfrak{su}(2)_\mathbb{C} \oplus \mathfrak{su}(2)_\mathbb{C}$ imply the existence of an isomorphism between the corresponding groups and, also, representations.  I.e., having found every irrep. of $SL(2,\mathbb{C})$, we have found every irrep of $SO(1,3)$.
6) We then use the Weyl unitarian trick to show that every representation of the group corresponding to $\mathfrak{su}(2) \oplus \mathfrak{su}(2)$ is completely reducible (we could also have evoked a similar result directly for semisimple Lie Algebras, which is the case of $\mathfrak{so}(1,3)$), since it is compact and, by isomorphism, extend this property to $SO(1,3)$, ensuring that we have found every representation of it.
Besides being possibly poorly phrased and way imprecise, does the above make sense? What are the misconceptions? How to fix the line of thought above while keeping the same language/strategy?
 A: So there is some confusion here both between Lie group and Lie algebra properties and also about passing back and forward between the complexification and real forms.
First things first, as Torsten says, simply connected is a property that is important for Lie groups not for Lie algebras. If we want to get technical, any Lie algebra is simply connected since it is just a vector space. The importance of simply connectedness in Lie theory is that a Lie algebra has several Lie groups corresponding to it. There is only one among these groups which is simply connected and crucially for every representation of the Lie algebra there is a corresponding representation of the simply connected groups. Others of these groups will also have some of these representations (they will all have an adjoint representation for example) but the simply connected one guarantees a one-to-one correspondence.
So talking about Lie algebra representations is "equivalent" to talking about simply-connected Lie group representations.
So then as you say you can learn about the representations of $\mathfrak{so}(3,1)$ by looking at its complexification $\mathfrak{so}(3,1)_{\mathbb{C}}=\mathfrak{so}_4 = \mathfrak{sl}_2\oplus \mathfrak{sl}_2$. Now, since you are a physicist, we should be careful about different naming conventions here. For every representation of a complex Lie algebra $\mathfrak{g}$ on a complex vector space $V$ there is a corresponding representation of each of real form $\mathfrak{g}_{\mathbb{R}}$ on the same complex vector space $V$ (just restrict the representation) and going the other way is similarly easy. Note these are not what physicists call real representations (but if we can find a real subspace $V_{\mathbb{R}} \leq V$ preserved by the representation we can make one of those). So there is a one-to-one correspondence again between (complex) representations of $\mathfrak{g}$ and of $\mathfrak{g}_{\mathbb{R}}$.
Note carefully that this does not give us a one-to-one correspondence of representations between representations of $SO(3,1)$ and $SL(2,\mathbb{C}) \cong Spin(3,1)$. Their Lie algebras are isomorphic (viewing the second one as a real group/algebra) but only one of them is simply connected ($SL(2,\mathbb{C})$). So every representation of $SO(3,1)$ corresponds to one of $SL(2,\mathbb{C})$ but not the other way round. The simplest examples of these missing representations are the spin representations. Just as in the definite case ($SO(4)$) there are two half-spin representations here.
