Lagrangian in $D$-dimensions in maximal supergravity In the paper Dualisation of Dualities. I. By Cremmer,Julia, Lu and Pope they derive the $D$-dimensional lagrangian in terms of the generators $E_{i}^j$, $E_{ijk}$ and $D$, I believe $E_{i}^j$ is the matrix with 1 in the $(i,j)$ position, but it is not specified what $E_{ijk}$ and $D$ are. My question is is it known what matrices they are? And if they are different for each dimension, how can they be derived? Is there more similar literature on the topic?
 A: The scalars in $D$ dimensions consist of 


*

*$(11-D)$ dilatons: $\phi_i, \ i = 1 \cdots 11-D$

*Axions $A_{(0)ijk}$ (call them Type A axions for now)

*Axions $A^i_{(0)j}$ (call them Type B axions for now)


If you go through the appendix in a bit of detail, you see that the dilaton vectors $\vec b_{ij}$ corresponding to Type B axions and $\vec a_{ijk}$ corresponding to Type A axions can be expressed as the positive root vectors of $E_{11-D}$ (up to dualization) i.e. to positive root vectors of $E_6, \ E_7$ or $E_8$. 
The roots associated to $\vec b_{ij}$ are $E^i_{j}$ and the roots associated to $\vec a_{ijk}$ are  $E^{ijk}$. 
So it answer your question: The matrices $E$ are associated with postive roots of one of the three exceptional groups and they do depend on the dimension. 
The $D$ that is sort of confusing here is an extra axion that appears in in certain dimensions, and therefore there is an extra generator. They call this also $D$, which can be confusing. Again, this depends on dimension. 
Regarding the derivation: What would you liked to have derived?
Regarding the literature survey: What specific topic would you do you want to read more about? The construction of scalar multiplets in D dimensional supergravity? Exceptional supergravity 
I'm happy to edit my answer accordingly. 
