*In what follows I'm going to use $V\_{\theta\_s}$ for the little adjoint representation af a Lie algebra i.e. the representation associated with the highest short rooth $\theta\_s$.*


Is easy to see that simple algebras of types $B\_n$, $C\_n$ and $F\_4$ can be found as subalgebras of respectively $D\_{n+1}$ with $n>3$, $A\_{2n-1}$ and $E\_6$ looking at the fixed poits algebra of an opportune involution (induced by an automorphism of the Dynkin Diagram)


In this article <http://arxiv.org/pdf/math/0303222.pdf> the autor says that if $\mathfrak{g}= \mathfrak{g}\_0\oplus \mathfrak{g}\_1$ is the decompoisition induced by the involution as above, then $\mathfrak{g}\_1$ is isomorphic to the $\mathfrak{g}\_0$-module $V(\theta\_s)$.


Looking at the article this seems to be very esasy to prove but I have no ideas and no reference.


Any help is going to be well accepted.


Probably this question is very easy for people on this site, in such a case forgive me, please.



Involutions and Little Adjoint Representations of Simple Algebras
The involutive compact Lie algebras yield symmetric spaces and the representations in point are the corresponding isotropy representations. 


The first symmetric space is the sphere $SO\_{2n+2}/SO\_{2n+1}=S^{2n+1}$, whose isotropy representation is the standard action of $SO\_{2n+1}$ on $\mathbb R^{2n+1}$. Its highest weight (after complexifying) is $\epsilon\_1$, where we denote the roots of $SO\_{2n+1}$ by $\pm\epsilon\_i\pm\epsilon\_j$ (long) and $\pm\epsilon\_i$ (short), hence, the highest short root. 


The second one is $SU\_{2n}/Sp\_n$ with isotropy representation $\Lambda^2\mathbb C^{2n}\ominus\mathbb C$ and highest weight $\epsilon\_1+\epsilon\_2$, where the roots are denoted $\pm\epsilon\_i\pm\epsilon\_j$ (short) and $\pm2\epsilon\_i$ (long), hence again the highest short root. 


The last one is $E\_6/F\_4$ with isotropy representation given by the irreducible 26-dimensional representation of $F\_4$. The roots of $F\_4$
are $\pm\epsilon\_i\pm\epsilon\_j$ (long; there are 24 of them) and $\pm\epsilon\_i$ (short; 8 roots) and $\frac12(\pm\epsilon\_1\pm\epsilon\_2\pm\epsilon\_3\pm\epsilon\_4)$ (short; 16 roots). The highest short root is $\epsilon\_1$ , and this is also the highest weight of the considered representation. 


About references: you can find the isotropy representations in Joseph Wolf's book *Spaces of constant curvature*, ch.8. For $F\_4$, you find complementary information in J. F. Adams' *Lectures on exceptional Lie groups* (Edited by Zafer Mahmud and Mamoru Mimura) and 
in the planches at the end of Bourbaki's book on Lie groups, ch. IV to VI. 


There could exist some conceptual argument, but I am not sure the work is worth for three representations only. In any case, I will write again if anything occurs to me. 


