Representations and Irreducibility. Representations of lie groups and algebras, sub-representations, irreducible representations, common operations (direct sum, tensor product, etc), intertwining operators, and proof of Schur's lemma.
Killing form, solvable, nilpotent and semi-simple lie algebras, existence of maximal solvable and nilpotent radical, equivalent definitions of semisimple lie algebras (in terms of killing form and direct sum of simple lie algebra).
References: Page 88-90 and 95-98 of the book "Lie Groups: An approach through invariants and representations" by Claudio Procesi.
The Lie-Kolchin theorem and Borel's Fixed Point Theorem
Complexification and Real Forms
References: For background on algebraic groups: Malle-Testermann's Linear Algebraic Groups and Finite Groups of Lie Type (Chapters 1 and 6), or Milne's Algebraic Groups, available here. For real forms I used Onishchik-Vinberg's Lie Groups and Lie Algebras III.