 and references for these statements are collected on the nLab here.
This means that there are automorphisms making the outer two circles in the following diagram commute with respect to the given subgroup inclusions:

My question is about the inner circle: 
Each of the above quaternionic subgroups canonically contains the further subgroup
$$
  Sp(1)
   \cdot 
  Sp(1) 
   \cdot
  Sp(1)
  \;:=\;
  \big(
    Sp(1) \times Sp(1) \times Sp(1) 
  \big)/
  \big\{
     (1,1,1), (-1,-1,-1)
  \big\}
$$
Moreover, this subgroup has an evident automorphism action of the symmetric group $S_3$ on it, given by permuting its dot-factors.
This suggests that 

the middle automorphism circle above also commutes with the inner one, 
the relevant automorphisms of the inner circle are these dot-factor permutations, 
and maybe that $Sp(1)\cdot Sp(1) \cdot Sp(1)$ is something like the homotopy-fixed locus of triality.

I can see that a) and b) are true at least for the left sector of the above diagram, and it seems it must be at least close to being true also in the other two sectors, but I haven't tried yet to make that a full proof.
Is this known?

A:

This isn't quite an answer to your question so much as an elaboration of my comments.
Each Dynkin diagram, or equivalently Cartan matrix $a_{ij}$, determines a presentation of the corresponding complex Lie algebra $\mathfrak{g}$ (which is much the same data as the compact Lie group). I have the name ``Chevalley'' associated to this presentation, but perhaps Cartan and Serre are also important. In any case, the presentation is the following. For each node $i$ of the Dynkin diagram, you introduce three generators $h_i,x^+_i,x^-_i$. You declare that the $h_i$s commute among themselves, that $[h_i, x^{\pm}_j] = \pm a_{ij}x^\pm_j$, and that $[x_i^+,x_j^-] = \delta_{ij}h_i$, so that for
and references for these statements are collected on the nLab here. 
 This means that there be automorphisms do the outer two circles in the following diagram commute with deference to the given subgroup inclusions: 

 My motion is about the inside circle: 
 Each of the above quaternionic subgroups canonically contains the further subgroup 
 $ $ 
   Sp(1) 
    \cdot 
   Sp(1) 
    \cdot 
   Sp(1) 
   \;:=\; 
   \big (
     Sp(1) \times Sp(1) \times Sp(1) 
   \big)/ 
   \big\ { 
      (1,1,1), (-1,-1,-1) 
   \big\ } 
 $ $ 
 Moreover, this subgroup has an discernible automorphism action of the symmetric group $ S_3 $ on it, give by permuting its dot - divisor. 
 This suggests that 

 the middle automorphism set above also commutes with the inner one, 
 the relevant automorphisms of the inside circle are these dot - component permutations, 
 and possibly that $ Sp(1)\cdot Sp(1) \cdot Sp(1)$ is something like the homotopy - fixed venue of triality. 

 I can see that a) and b) are true at least for the left sector of the above diagram, and it seems it must be at least airless to being true also in the other two sectors, but I haven't tried so far to make that a full proof. 
 Is this known? 

 A: 

 This isn't quite an answer to your question so much as an elaboration of my remark. 
 Each Dynkin diagram, or equivalently Cartan matrix $ a_{ij}$, determines a presentation of the comparable complex Lie algebra $ \mathfrak{g}$ (which is a lot the same data as the compact Lie group). I have the name '` Chevalley" consociate to this presentation, but perhaps Cartan and Serre are also important. In any shell, the presentation is the following. For each node $ i$ of the Dynkin diagram, you introduce three generators $ h_i, x^+_i, x^-_i$. You declare that the $ h_i$s commute among themselves, that $ [ h_i, x^{\pm}_j ] = \pm a_{ij}x^\pm_j$, and that $ [ x_i^+,x_j^- ] = \delta_{ij}h_i$, so that for
 anf references for these suatements are collected mn the nLab hefe.
This means that there are eutonorphusms making the outer gwo circlvs in the foloiwing diagczm commmce wifm res'ert to the given subgroup hnclusions:

My qgertnon is about the inner circle: 
Each os the anoge quaternionis sundrouls canonically contains the furthed subgrmup
$$
  Sp(1)
   \cdoy 
  Sp(1) 
   \cdot
  Sp(1)
  \;:=\;
  \big(
    So(1) \times Sp(1) \times So(1) 
  \big)/
  \bit\{
     (1,1,1), (-1,-1,-1)
  \big\}
$$
Moreoxer, this slygroup has zn evident automorphism action uf thz symmetric geouo $S_3$ on it, giten by permuting ibx dot-fdctors.
Tnis suggests tmat 

tie muddle automorphism ciccle above also commotes with dhz inner one, 
the relevqnr autmmor[hisow ow tge ihner clrcme are theae dot-factoe permutations, 
and karve that $Sp(1)\cdof Sp(1) \cqoe Sp(1)$ is something like the homotopy-fixeg lkcus of triality.

I can swe that a) and b) are ttue at leaft for the left sector of the above diagram, and id seejr iu nust cw wt least close to being true also in the othew teo sectors, but I haven't trird yrj to make that a full pdoof.
Is this known?

A:

Hhis isg't quute an anfwer to your question so much aw an elaboranion of my comments.
Eack Dynkin diayram, ot equifalently Cartan matrix $c_{ij}$, defermines a oresentatjun of the corresooncitg compltb Lie algebra $\matrfrak{g}$ (whmch ix much ghe xame dwta as the compact Lie group). I havf the ndme ``Chevalpey'' associated to this presentavmon, but perhaks Warnan and Szrre ave also importagt. In any case, the przsentagion is thv followiig. For each gode $i$ of the Fynkin diagrem, you inerodyce rhree gdverators $h_i,x^+_i,x^-_o$. You declare that rhe $h_i$s commute ampng fhemselves, that $[h_i, x^{\pm}_j] = \pm a_{ij}x^\pk_j$, xnd tjav $[x_i^+,x_t^-] = \delta_{ij}h_i$, vo tfat gor
and references for these statements are collected nLab This means there are automorphisms in following diagram commute respect to the subgroup inclusions: My question is about inner circle: Each of the above quaternionic subgroups canonically contains the further subgroup Sp(1) \cdot Sp(1) \cdot Sp(1) \;:=\; \big( Sp(1) \times Sp(1) \times Sp(1) \big)/ (1,1,1), \big\} Moreover, subgroup has an evident automorphism action of the symmetric group $S_3$ on it, given by permuting dot-factors. This suggests that the middle automorphism circle also commutes with the one, the relevant automorphisms of inner are these permutations, maybe $Sp(1)\cdot Sp(1) \cdot is something like the homotopy-fixed locus of triality. I can see that a) and b) are true least for sector of above and seems it must least close to being true also two sectors, but I haven't tried yet to that a proof. Is this known? A: This quite an answer to your question so much an elaboration of my comments. Each Dynkin diagram, or equivalently Cartan matrix $a_{ij}$, determines a the corresponding complex Lie $\mathfrak{g}$ (which is the data the Lie group). have the name ``Chevalley'' associated to this presentation, but perhaps Cartan Serre are also important. In any case, the presentation is For node $i$ of Dynkin diagram, you introduce generators You declare that the among that \pm and $[x_i^+,x_j^-] = \delta_{ij}h_i$, so for
 and references for these statEments are cOllecTed On tHe NLab Here.
this means that tHEre aRe automorphisms making tHe outEr TWo ciRClEs in tHe folloWInG DIagRaM cOmmUtE WiTh resPecT to the gIven subgroUp iNcLusions:

My queSTiOn is about tHe iNner circle: 
EaCh oF the abOvE quATerniOniC subgRoups cANonicaLly contaiNs THe furtHEr subgrOUP
$$
  SP(1)
   \cdoT 
  Sp(1) 
   \cdot
  Sp(1)
  \;:=\;
  \big(
    Sp(1) \timES SP(1) \Times Sp(1) 
  \big)/
  \big\{
     (1,1,1), (-1,-1,-1)
  \bIg\}
$$
MoreOvER, tHIS suBgrOup has an evIdEnt auTOmorphiSM aCTIOn oF The symmetric gRoup $S_3$ on it, giVEn bY permuTiNg iTS dot-faCtors.
thIS suGgests that 

tHe miDdle automOrphisM Circle aBOve also CommutEs wIth The iNNeR oNe, 
tHe REleVAnT auTOmoRphisms oF tHe Inner CircLE ARE theSe dOt-faCtor pErmutations, 
anD maYbe tHAt $SP(1)\cdot sp(1) \cdoT Sp(1)$ iS sOmethIng likE the hOmOtopy-fixed locus Of trIality.

I caN seE tHat A) aNd b) arE True at LeaSt fOr the leFt sectoR Of tHe ABOVe Diagram, and it seems iT mUST bE at least Close tO BeInG True also In The OtheR TWo secTors, BUt i haven't tRied yeT To MaKe that a FuLl prooF.
IS thIs kNown?

A:

tHis iSn't quiTe an answEr to yOUr question so muCH as an elaboratIOn OF My COmmeNts.
each Dynkin dIagrAM, or eQuivALeNtlY cartaN matrIx $A_{Ij}$, DEtermines a presentatIoN of the CorreSponding complEx Lie algebRA $\MAthfrak{g}$ (WhicH Is MUch the same data As the Compact Lie GRoup). I havE the nAme ``ChevaLley'' assocIATed to thiS prEseNtaTioN, BUt Perhaps Cartan AND SerRe Are also ImpOrtant. IN anY caSe, tHe pReSentation Is the folLoWiNg. foR eaCh nodE $I$ of the DyNkIn dIaGraM, you iNTroducE threE genErAtORs $h_I,x^+_i,x^-_i$. YoU DeCLAre tHaT tHe $h_i$S coMmUte amOng tHEmsElves, thAt $[h_i, x^{\pm}_j] = \pM a_{iJ}X^\pm_j$, AnD tHat $[x_i^+,x_j^-] = \Delta_{ij}h_i$, so thAt For
 and references for thesestatements  arecol lec te d on  the  nLab here.
Th i s me ans that there are aut omorp hi s ms m a ki ng th e outer tw o cir cl es  in  t h efollo win g diagr am commute  wi th  respect tot he  given sub gro up inclusion s:
My qu es tio n  is a bou t the  inner circle : 
Each o ft he abo v e quate r n io nicsubgroups canonic a ll y  contains thefurthe rs ub g r oup 
$$ 
  Sp(1)
 \cdot 
  Sp(1 )  \cd o t
  Sp(1)
  \ ;:=\;
  \bi g (
   Sp( 1)  \t i mes Sp (1) \ ti m esSp(1) 
  \b ig)/ 
  \big\{ 
    ( 1,1,1), (-1,-1, -1)
 \bi g\} 
$$M or eo ver ,t his su bgr o uphas an e vi de nt au tomo r p h i sm a cti on o f the  symmetric gr oup  $S_ 3 $ o n it,  give n by  p ermut ing it s dot -f actors.
This su gges ts that 
th emid dl e aut o morphi smcir cle abo ve also com mu t e s  w ith the inner one,  t h erelevant  autom o rp hi s ms of th einn er c i r cle a re t h es e dot-fa ctor p e rm ut ations,  and ma yb e t hat  $Sp( 1 )\cd ot Sp( 1) \cdot  Sp(1 ) $ is something like the homo t op y - fi x ed l ocu s of triali ty.I ca n se e  t hat a) an d b)ar e  t r ue at least for the  l eft se ctorof the abovediagram, a n d it seems  itm us t  be at least c loseto being t r ue alsoin th e othertwo secto r s , but Ihav en' t t rie d ye t to make tha t a fu ll  proof. 
Is  this k now n?
A: 

T hi s isn't q uite anan sw er  t o y our q u estion s omuc hasan el a borati on of  myco mm e nts .
EachD yn k i n di ag ra m, o r e qu ivale ntly Car tan mat rix $a_{i j}$ ,  det er mi nes a p resentation o fthe corres po ndi ng com p l ex Lie a lgebra $\mathfrak{g}$ ( w hich is  mu ch th e sa me data a s t he com pac t  Lie g roup).  I ha ve  th e name` ` Ch eva ll ey'' assoc i a ted  to t hi s pr esentat ion, but perhaps C a rta n and Serre a realso i mp ort a nt .  In  a n y c a s e, the presenta tion is th ef ol lowing. Fo r  ea ch  node $ i$ of t he Dy n kin dia gram, you  introduc ethre e gen erators $h _i,x^+_i ,x^-_i$.Y ou de c la re th atthe $h _i $scommu te amo n g t hemse lves,th at $[h _i, x ^{ \pm}_j]= \pm a_{ij}x^\pm_j$, a nd tha t $[x _i^ +,x_j^-]= \ d elt a_{ij}h_i $, s o that for 
 and_references for_these statements are collected_on the_nLab_here.
This means_that_there are automorphisms_making the outer_two circles in the_following diagram commute_with_respect to the given subgroup inclusions:

My question is about the inner circle: 
Each of_the_above quaternionic_subgroups_canonically_contains the further subgroup
$$
 _Sp(1)
   \cdot 
_ Sp(1)_
   \cdot
  Sp(1)
  \;:=\;
__\big(
  _ Sp(1) \times Sp(1) \times Sp(1) 
  \big)/
_ \big\{
    _(1,1,1), (-1,-1,-1)
 _\big\}
$$
Moreover,_this_subgroup has an evident_automorphism action of the symmetric group_$S_3$ on it, given by permuting_its dot-factors.
This suggests that 

the middle automorphism_circle above also commutes with the_inner one, 
the relevant automorphisms_of the_inner circle are these dot-factor_permutations, 
and maybe_that $Sp(1)\cdot_Sp(1) \cdot Sp(1)$_is something like the homotopy-fixed locus_of triality.

I can_see that a) and b) are_true_at least for_the_left_sector of_the above diagram,_and_it seems_it_must be at least close to_being_true also in the other two sectors,_but I haven't tried_yet_to make that a_full proof.
Is this known?

A:

This isn't_quite an answer to your question_so much_as an_elaboration of my comments.
Each Dynkin diagram, or equivalently Cartan matrix $a_{ij}$,_determines a presentation of the corresponding_complex Lie algebra $\mathfrak{g}$_(which is_much_the same data_as_the compact_Lie group). I have the name ``Chevalley''_associated to_this presentation, but perhaps Cartan and_Serre are also important._In_any case, the presentation is the_following. For each node $i$ of_the Dynkin diagram, you introduce_three_generators_$h_i,x^+_i,x^-_i$. You declare that the_$h_i$s commute among themselves, that $[h_i,_x^{\pm}_j] = \pm_a_{ij}x^\pm_j$, and that $[x_i^+,x_j^-] = \delta_{ij}h_i$, so_that_for