Quaternions, Lie Groups and Lie Algebras. Steps to realize a paper. I have to realize a paper about quaternions and Lie Groups and Lie Algebras.  How can I realize the links between quaternions and Lie Groups & Algebras. Which books do you recommend me? First, I have to read about quaternions or Lie Group and Lie Algebra? 
Thanks in advanced for your help :) 
 A: I understood that you are not completely familiar with Lie Groups and Lie Algebras. From the top of my head I could list three connections (see bellow) but I do not think you can do much without having some basic knowledge on Lie Algebras and Lie Groups:


*

*The study of the complex representations of compact Lie Groups can be done by classifying the irreducible representations as real (which are the ones obtained by complexification), quaternionic (which are "representations" of the group in some $\mathbb H^n$). A reference for this subject is the book: "Representations of Compact Lie Groups" by T. Bröcker and T.tom Dieck

*The Lie Group $SO(3)$ can be realized as the the group of automorphisms of $\mathbb H$. Also, the Lie Algebra $\mathfrak so(4)$ can be realized as the algebra of the derivations in $\mathbb H$. You can find references of these fact in lots of pdf over the internet (as I remmeber).

*You can define a Lie bracket in $\mathfrak{sl}(\operatorname{Im}(\mathbb H))\times (\operatorname{Im}(H))^2$ so that it is a realization of the normal real form of the simple Lie Algebra $G_2$. The only references I have for this are in Portuguese, for example my master's dissertation: http://www.bibliotecadigital.unicamp.br/document/?code=000893270
Since I feel that your plans are very broad, there is the great book which relates the normed algebras (as the quaternionionic algebra) with the exceptional Lie Algebras (look for the classification of the simple Lie Algebras in wikipedia): "Octonions, Jordan Algebras, and Exceptional Groups" by Tonny A. Springer, Ferdinand D. Veldkamp.
I hope this helps on something.
