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I would ask three questions, but none of these questions are meant to be particular difficult to solve so I figured it would be a waste of space to post three separate threads.

(1) Suppose I have a Lie algebra $L$ (semisimple, complex) corresponding to a root system $\Phi$, say $B_2$ for example. This root system has basis $\{ \alpha, \beta \}$.

I would like to calculate the dimension of $L$ and the dimension of the Cartan subalgebra $H$ from this root system, however I'm a bit unsure how to proceed; this is a low mark question from a past paper so I assume it's fairly trivial and I'm missing something obvious.

We have not used the Coxeter number in our course so I'd like to calculate these without involving it (the previous question on this topic posted on Stackexchange involves this number).

(2) Another question I have is showing the dual space $H^*$ of the Cartan subalgebra = $<\Phi>_\mathbb{C}$ (for general $\Phi$ set of roots of $H$), I know $\forall h \in H, \exists \alpha \in \Phi$ such that $\alpha(h) \neq 0$ so for any $\beta \in H^*$ we can represent $\beta(h) = \lambda_h \alpha(h)$ for some $\lambda_h \in \mathbb{C}$, $\alpha \in \Phi$; however I get stuck trying to extend this to showing $\beta$ is a linear combination of $\alpha_1, \dotsc, \alpha_n \in \Phi$.

(3) Finally for $L = L_1 \oplus \dots \oplus L_k$ with $L_i$ simple ideals of $L$ for $1 \leq i \leq k$, I would like to show that $H \cap L_i$ is a Cartan subalgebra of $L_i$ given that $H$ is a Cartan subalgebra of $L$. Proving the abelian, semisimple and subalgebra criteria are fairly trivial but I'm struggling to show that $H \cap L_i$ not being maximal gives us a contradiction.

Thanks for any helps/hints on any of the questions, I'm fairly sure the details I'm missing on all three are simple things since the questions are meant to be solved very quickly.

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    $\begingroup$ $(1)$ We have $dim L=|\Phi|+rank(L)$. $\endgroup$ – Dietrich Burde May 18 '16 at 17:48

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