I would like to know about the literature concerning the group of outer automorphisms of the Lie algebra $\mathfrak{sl}(2,R)$. This question is addressed in different places in a contradictory way. In certain works, e.g. 

M.A. Farinati and A.P Jancsa,
Three dimensional real Lie bialgebras,
Revista de la union matematica argentin Vol. 56, No. 1, 2015, Pages 27–62, 2015,

it is claimed the the aforesaid group is trivial and all automorphisms of SL(2,R) are inner. By googling the question, I also found several sources claiming that the group of inner automorphisms of $\mathfrak{sl}(2,R)$ is PSL(2,R) (see https://groupprops.subwiki.org/wiki/Special_linear_group:SL(2,R)), and the outer automorphisms are given by PGL(2,R) (see https://mathoverflow.net/questions/14735/outer-automorphisms-of-simple-lie-algebras) which are obviously different.

Similarly, I would like to know about the structure of the group of outer automorphisms of the Lie algebra $\mathfrak{su}(2)$

Thank you in advance for your comments.