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I have been trying to learn about the classification of semi-simple Lie algebras and their representations using the Cartan Weyl approach i.e. Cartan subalgebra, roots, weights...

I would like to find a book that includes proofs since most times I come across statements and formulas I just have to accept. Some examples of the proofs I would like to see are about:

How do I know that the algebra contains elements $H_i$ such that $ad_{H_i}$ are diagonalizable maps?

Why $[H_i,H_j]=0$ implies that $ad_{H_i}$ and $ad_{H_j}$ are simultaneously diagonalizable?

How do I know that the eigenvectors of $ad_{H_i}$ are a basis of the algebra?

How to derive the properties of the roots? For example, if $\alpha$ is a root then the only multiples which are roots are $\pm\alpha$.

How to derive the properties of the Cartan matrix.

I would appreciate any recommendations where I can read about this topics. Thank you in advance!

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    I do recommand Fulton and Harris book. They explain it really clearly and present in detail the representation of $\mathfrak{sl}_2$ which is the beginning of everything. They have a detailed appendix when they explain many things as well. With a standard background in linear algebra you can read it. – 2017-01-03
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    If you search MSE here you will find an answer for *all* of your questions. One example: see [here](http://math.stackexchange.com/questions/236212/simultaneously-diagonalizable-proof) for the answer why $ad_{H_i}$ and $ad_{H_j}$ are simultaneously diagonalizable. For Cartan subalgebra see [here](http://math.stackexchange.com/questions/1684629/cartan-subalgebra-and-the-decomposition-of-eigenspaces) etc. – 2017-01-03
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    Erdmann and Wildon's book http://www.springer.com/gb/book/9781846280405 is very self-contained and has the classification. – 2017-01-03
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    There is also the book by Humphreys. Which books have you been reading that did not include the proofs? I have also written a "guide" to the classification at http://math.stackexchange.com/questions/427135/relation-between-root-systems-and-representations-of-complex-semisimple-lie-alge/435135#435135 – 2017-01-03

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