I am reading Lascar's article "Omega-stable groups" in the book of Bouscaren "Model Theory and Algebraic Geometry". In section 5, he applies the indecomposability theorem to show that If $G$ is a connected $\omega$-stable group of finite Morley rank, then its derived subgroup $[G,G]$ is definable and connected. There are some points in the proof that I can not make it clear.
He defines $Y_a=\{{ba^{-1}b:b\in G\}}$ and claims that it is indecomposable. However, I don't understand why this set is definable (before going to prove the decomposability).
He used the fact that if $G$ is connected and $H$ is a normal definable subgroup then $G/H$ is connected. I don't know how to prove it. The obstruction is that $G/H$ might not be definable, and hence if we take $N$ to be a definable subgroup of finite index of $G/H$, then its preimage in $G$ might not be definable in $G$ (although it is of finite index).
He also mentioned that $[G,G]$ is always definable without the assumption $G$ being connected (without proof). To prove it, I write $$G=\bigsqcup_{g_i\in I} g_iG, a=g_ia', b=g_jb'$$ and then $$[G,G]=<g_i Y_{g_jb'}g_i^{-1}b'^{-1}g_j^{-1}:g_i,g_j\in I, b'\in G^0>, Y_{g_jb'}=\{a'g_jb'a'^{-1}:a'\in G^0\}$$ where $G^0$ is the connected component of $G$. So, similar to Lascar's proof, for each $g_i,g_j,b'$, the set $Y_{g_jb'}$ is indecomposable, hence $[G,G]$ is connected and definable. However, it seems that $[G,G]$ is not always connected (for example in the case of algebraic groups). I don't understand what is the problem in this argument.
I believe I am missing some points involving definability, especially the imaginaries, right?