THE TORSION OF THE $\G_2$-CONNECTION $\NABLA^N$ IS A 3-FORM IF AND ONLY IF $A=B$. IN THIS CASE IT IS GIVEN BY A MULTIPLE OF THE $\G_2$-FORM,
\[
T= \FRAC{-1}{\SQRT{30A}}\ \OMEGA_{\PSI_0} ,
\]
AND THUS IN PARTICULAR IS INVARIANT AND $\NABLA^N$-PARALLEL.
\END{PROPOSITION}
\BEGIN{PROOF}
FROM $\NABLA^N =\NABLA^G -\GAMMA$ WE SEE THAT THE DIFFERENCE TENSOR BETWEEN $\NABLA^N$ AND $\NABLA^{G_{A,B}}$ IS $-\GAMMA = \FRAC{2}{3}A\LRCORNER \OMEGA_{\PSI_0}$, AND THUS THE TORSION IS A 3-FORM IF AND ONLY IF $(X,Y,Z)\MAPSTO \FRAC{2}{3}\OMEGA_{\PSI_0}(A(X),Y,Z)$ IS TOTALLY SKEW-SYMMETRIC. CLEARLY THIS HAPPENS PRECISELY WHEN $A$ IS A MULTIPLE OF THE IDENTITY, WHICH OCCURS IF AND ONLY IF $A=B$. WHEN $A=B$ WE HAVE $A=\FRAC{-\SQRT{3}}{4\SQRT{10A}} \ID $ AND THUS THE TORSION OF $\NABLA^N$ IS GIVEN BY $$T = -2\GAMMA  = -2(-\FRAC{2}{3}A\LRCORNER \OMEGA_{\PSI_0}) = -\FRAC{1}{\SQRT{30A}} \OMEGA_{\PSI_0}. $$
\END{PROOF}
\BEGIN{REMARK}
FROM [22] IT IS KNOWN THAT THE CHARACTERISTIC CONNECTION $\NABLA^C$ (WHICH EXISTS IN THIS EXAMPLE SINCE THE $\G_2$-STRUCTURE IS COCALIBRATED) IS UNIQUE, SO WHEN $A=B$ IT COINCIDES WITH $\NABLA^N$. WHEN $A\NEQ B$ WE NEED TO FIND A DIFFERENT WAY TO DESCRIBE $\NABLA^C$. USING THEOREM 4.8 IN [22], AND TAKING INTO ACCOUNT THAT OUR $\G_2$-STRUCTURE IS COCALIBRATED AND DIFFERS FROM THE REFERENCE $\G_2$-FORM IN [22] BY AN ORIENTATION-REVERSING CHANGE OF BASIS, THE TORSION 3-FORM OF $\NABLA^C$ IS GIVEN BY
\BEGIN{ALIGN}
T^C &= -\FRAC{1}{6}\LANGLE D\OMEGA_{\PSI_0},\AST\OMEGA_{\PSI_0}\RANGLE \OMEGA_{\PSI_0} + \AST D\OMEGA_{\PSI_0} . \LABEL{CHARTORSIONFORM}
\END{ALIGN}
\END{REMARK}
%FOR THE FINAL CLAIM IN THE FOLLOWING LEMMA, RECALL (E.G. FROM TABLE 6 IN [6]) THAT THE $\G_2$-STRUCTURE DETERMINED BY THE $3$-FORM $\OMEGA_{\PSI_0}$ IS NEARLY PARALLEL IF AND ONLY IF $D\OMEGA_{\PSI_0}$ IS A NON-ZERO SCALAR MULTIPLE OF $\AST \OMEGA_{\PSI_0}$.
IN ORDER TO COMPUTE $T^C$ USING (\REF{CHARTORSIONFORM}) WE FIRST PROVE THE FOLLOWING LEMMA:
\BEGIN{LEMMA}
THE HODGE DUAL AND EXTERIOR DERIVATIVE OF $\OMEGA_{\PSI_0}$ ARE GIVEN BY
\BEGIN{ENUMERATE}[(I)]
\ITEM $\AST\OMEGA_{\PSI_0} = -E_{4567} +E_{2367} +E_{2345} +E_{1357} -E_{1346} +E_{1256} +E_{1247},  $
\ITEM $ D\OMEGA_{\PSI_0} = \FRAC{A+5B}{B\SQRT{30A}} (\AST \OMEGA_{\PSI_0}) + \FRAC{-5A+5B}{B\SQRT{30A}} E_{4567}.  $
\END{ENUMERATE}
%IN PARTICULAR, THE $\G_2$-STRUCTURE INDUCED BY $\OMEGA_{\PSI_0}$ IS NEARLY PARALLEL IF AND ONLY IF $A=B$.
\END{LEMMA}
\BEGIN{PROOF}
THE FIRST CLAIM IS STRAIGHTFORWARD. THE PROOF OF THE SECOND CLAIM PROCEEDS BY A LENGTHY YET STRAIGHTFORWARD COMPUTATION USING THE ABOVE EXPRESSION OF THE NOMIZU MAP $\UPLAMBDA^{G_{A,B}} $ TOGETHER WITH THE IDENTITY $D = \SUM_{I=1}^7 E_I\WEDGE \NABLA_{E_I}^{G_{A,B}} $.
\END{PROOF}
RECALLING (E.G. FROM \CITE[TABLE 6]{SRNI}) THAT A $\G_2$-STRUCTURE $\OMEGA$ IS NEARLY PARALLEL IF AND ONLY IF $D\OMEGA$ IS A NON-ZERO SCALAR MULTIPLE OF $\AST \OMEGA$, THE PRECEDING LEMMA ALSO GIVES AN ALTERNATE PROOF OF THE FACT THAT THE $\G_2$-STRUCTURE INDUCED BY $\PSI_0$ IS NEARLY PARALLEL IF AND ONLY IF $A=B$.
\BEGIN{COROLLARY}
THE CHARACTERISTIC CONNECTION $\NABLA^C$ IS GIVEN BY
\BEGIN{ALIGN*}
\NABLA^C &= \NABLA^{G_{A,B}} -\FRAC{1}{2}\LEFT( \FRAC{A}{B\SQRT{30A}} \OMEGA_{\PSI_0} +\FRAC{5A-5B}{B\SQRT{30A}} E_{123}\RIGHT) ,
\END{ALIGN*}
AND THE DEFINING 3-FORM $\OMEGA_{\PSI}$ AND DEFINING SPINOR $\PSI_0$ ARE BOTH $\NABLA^C$-PARALLEL.
\END{COROLLARY}
\BEGIN{PROOF}
USING (\REF{CHARTORSIONFORM}) AND THE PRECEDING LEMMA, WE CALCULATE
\BEGIN{ALIGN*}
T^C &= -\FRAC{1}{6} \LANGLE \FRAC{A+5B}{B\SQRT{30A}} (\AST \OMEGA_{\PSI_0}) + \FRAC{-5A+5B}{B\SQRT{30A}} E_{4567}, \AST \OMEGA_{\PSI_0}\RANGLE \OMEGA_{\PSI_0} + \AST\LEFT( \FRAC{A+5B}{B\SQRT{30A}} (\AST \OMEGA_{\PSI_0}) + \FRAC{-5A+5B}{B\SQRT{30A}} E_{4567}\RIGHT)  \\
& = \LEFT( \FRAC{-7(A+5B)}{6B\SQRT{30A}} +   \FRAC{-5A+5B}{6B\SQRT{30A}} \RIGHT) \OMEGA_{\PSI_0}         +  \LEFT(  \FRAC{A+5B}{B\SQRT{30A}} \OMEGA_{\PSI_0} +\FRAC{-5A+5B}{B\SQRT{30A}} E_{123}\RIGHT)  \\
&= -\FRAC{A}{B\SQRT{30A}} \OMEGA_{\PSI_0} -\FRAC{5A-5B}{B\SQRT{30A}} E_{123}.
\END{ALIGN*}
}THE FACT THAT $\NABLA^C\OMEGA = 0= \NABLA^C\PSI_0$ FOLLOWS SINCE $\NABLA^C$ IS A $\G_2$-CONNECTION.
\END{PROOF}
A NATURAL QUESTION IS WHETHER THERE ARE OTHER INVARIANT DIFFERENTIAL FORMS. TO THAT END, ONE FINDS:
\BEGIN{PROPOSITION}\LABEL{SP2SP1_INV_FORMS_STATEMENT}
UP TO TAKING HODGE DUALS, THE INVARIANT DIFFERENTIAL FORMS ON $(S^7=\FRAC{\SP(2)\SP(1)}{\SP(1)\SP(1)},G_{A,B})$ ARE GIVEN IN TABLE \REF{TAB:INV_FORMS_S7_SP2SP1}.
\END{PROPOSITION}
\BEGIN{PROOF}
BY HAND, OR USING THE LIE COMPUTER ALGEBRA PACKAGE ([36]), ONE FINDS THAT THE DIMENSIONS OF THE SPACES OF INVARIANT $1$-, $2$-, AND $3$-FORMS ON $\SP(2)/\SP(1)$ ARE $3$, $6$ AND $10$ RESPECTIVELY. BASES FOR THESE SPACES ARE GIVEN BY $\{ \ETA_I\}_{I=1}^3 $, $\{ \PHI_I\RVERT_{\MATHCAL{V}} , \PHI_I\RVERT_{\MATHCAL{H}}  \}_{I=1,2,3} $, AND $\{\ETA_1\WEDGE \ETA_2\WEDGE \ETA_3\} \CUP \{ \ETA_I\WEDGE \PHI_J\RVERT_{\MATHCAL{H}} \}_{I,J=1,2,3} $ RESPECTIVELY. THE RESULT THEN FOLLOWS BY CHECKING WHICH ELEMENTS IN THESE SPANS ARE INVARIANT UNDER THE ADDITIONAL THREE ISOTROPY OPERATORS DESCRIBED IN THE PROOF OF THEOREM \REF{EXAMPLE_INVARIANT_SPINORS_SP_TWISTED_7SPHERE}.
\END{PROOF}
\BEGIN{TABLE}[H!]
\CENTERING
\CAPTION{INVARIANT DIFFERENTIAL FORMS ON $(S^7=\FRAC{\SP(2)\SP(1)}{\SP(1)\SP(1)},G_{A,B})$}
\BEGIN{TABULAR}{|L||L|L|}\HLINE
$K$  & $\DIM \LAMBDA^K_{\INV}$ & BASIS FOR $ \LAMBDA^K_{\INV}$ \\\HLINE
$0$    & $1$ & $1$  \\
$1$ & $0$ & $0$  \\
$2$ & $0$ & $0$ \\
$3$   & $2$ & $\OMEGA_{\PSI_0}$, $\XI_{1,2,3}$   \\ \HLINE
\END{TABULAR}
\LABEL{TAB:INV_FORMS_S7_SP2SP1}
\END{TABLE}
NOTING FROM THE PRECEDING PROPOSITION THAT THERE ARE NO INVARIANT $1$-FORMS, ONE OBTAINS:
\BEGIN{COROLLARY}\LABEL{S7_SP2SP1_NO_INV_SASAKIAN_STRUCTURE}
THE SPACE $(S^7 = \FRAC{\SP(2)\SP(1)}{\SP(1)\SP(1)}, G_{A,B}) $ DOES NOT ADMIT AN INVARIANT EINSTEIN-SASAKIAN OR 3-SASAKIAN STRUCTURE FOR ANY VALUES OF $A,B>0$.
\END{COROLLARY}
%NOTE THAT ONLY $K\LEQ 3$ IS SHOWN IN THE ABOVE TABLE SINCE INVARIANT %FORMS OF DEGREE $K\GEQ 4$ ARE OBTAINED FROM THOSE ABOVE BY APPLYING %THE HODGE STAR OPERATOR.
\END{EXAMPLE}
\SUBSECTION{$S^1$-QUATERNIONIC SPHERES, $S^{4N-1} = \FRAC{\SP(N) \U(1)}{\SP(N-1) \U(1)}$}
THIS IS THE CASE FROM SECTION \REF{CASEIIK=U1} CORRESPONDING TO $K=(1)$. VIEWING $(1)$ AS A SUBGROUP OF $(1)$ VIA $E^IΘ↦COSΘ+ ISINΘ+0J+0K$, AND FROM (\REF{INCLUSION_ISOTROPY_PRODUCTGROUP}), WE HAVE AT THE LEVEL OF LIE ALGEBRAS
\BEGIN{ALIGN*}
\MATHFRAK{SP}(N)\OPLUS \MATHFRAK{U}(1)&= \SPAN_{\R} \{ (IF_{P,Q}^{(N)},0) , (JF_{P,Q}^{(N)},0) ,(KF_{P,Q}^{(N)},0) ,(E_{R,S}^{(N)},0), (0,I)     \}_{\SUBSTACK{1\LEQ P\LEQ Q \LEQ N \\ 1\LEQ R < S \LEQ N} } ,  \\
\MATHFRAK{SP}(N-1)\OPLUS \MATHFRAK{U}(1)&=  \SPAN_{\R} \{ (IF_{P,Q}^{(N)},0) , (JF_{P,Q}^{(N)},0) ,(KF_{P,Q}^{(N)},0) ,(E_{R,S}^{(N)},0), (IF_{1,1}^{(N)},I)     \}_{\SUBSTACK{2\LEQ P\LEQ Q \LEQ N \\ 2\LEQ R < S \LEQ N} }     .
\END{ALIGN*}
NOTE THAT THE KILLING FORM OF $𝔰𝔭(N)⊕𝔲(1)$ FAILS TO BE NON-DEGENERATE, SO IN ORDER TO CHOOSE A REDUCTIVE COMPLEMENT WE INSTEAD TAKE THE ORTHOGONAL COMPLEMENT WITH RESPECT TO THE RESTRICTION OF THE INNER PRODUCT $Κ$ FROM (\REF{SQUASHEDKF}) TO THE SUBALGEBRA $𝔰𝔭(N)⊕𝔲(1)⊂𝔰𝔭(N)⊕𝔰𝔭(1)$,
\BEGIN{ALIGN*}
\MATHFRAK{M}&:= (\MATHFRAK{SP}(N-1)\OPLUS \MATHFRAK{U}(1))^{\PERP_{\KAPPA}}.
\END{ALIGN*}
THE ISOTROPY REPRESENTATION SPLITS INTO ONE COPY OF THE TRIVIAL REPRESENTATION AND TWO NON-ISOMORPHIC IRREDUCIBLE REPRESENTATIONS:
\[\MATHFRAK{M}\SIMEQ \MATHFRAK{M}_1\OPLUS \MATHFRAK{M}_2\OPLUS \MATHFRAK{M}_3,\]
WHERE $_𝔪_1=1$, $_𝔪_2=2$, AND $_𝔪_3 = 4(N-1)$. THIS GIVES A 3-PARAMETER FAMILY OF INVARIANT METRICS,
\BEGIN{ALIGN*}
G_{A,B,C}&:= -A \KAPPA\RVERT_{\MATHFRAK{M}_1\TIMES \MATHFRAK{M}_1} - B \KAPPA\RVERT_{\MATHFRAK{M}_2\TIMES \MATHFRAK{M}_2} -C\KAPPA\RVERT_{\MATHFRAK{M}_3\TIMES \MATHFRAK{M}_3} , \QQUAD A,B,C>0. \END{ALIGN*}
IN PARTICULAR, BY MANUALLY CHECKING THE SECTIONAL CURVATURES, ONE FINDS THAT THE ROUND METRIC IS GIVEN BY THE PARAMETERS
\[
A=\FRAC{N+3}{8(N+1)}, \QUAD B= \FRAC{1}{4(N+1)}, \QUAD C= \FRAC{1}{8(N+1)}.
\]
FOR GENERAL $A,B,C>0$, A $G_A,B,C$-ORTHONORMAL BASIS FOR $𝔪 $ IS GIVEN BY
\BEGIN{ALIGN*}
\XI_1&:= \FRAC{1}{\OMEGA}\LEFT( IF_{1,1}, -\LEFT( \FRAC{N+1}{2}\RIGHT) I\RIGHT), \QUAD
\XI_2:=\FRAC{1}{2\SQRT{B(N+1)}}\LEFT( -KF_{1,1}, 0\RIGHT) ,\QUAD
\XI_3:=\FRAC{1}{2\SQRT{B(N+1)}}\LEFT(JF_{1,1}, 0\RIGHT) ,\\
E_{4P}&:= \FRAC{1}{2\SQRT{2C(N+1)}}(JF_{1,P+1} ,0),\QUAD
E_{4P+1}:=  \FRAC{1}{2\SQRT{2C(N+1)}}(KF_{1,P+1} ,0),\\
E_{4P+2}&:= \FRAC{1}{2\SQRT{2C(N+1)}}(IF_{1,P+1} ,0),\QUAD
E_{4P+3}:= \FRAC{1}{2\SQRT{2C(N+1)}}(E_{1,P+1} ,0),
\END{ALIGN*}
FOR $1≤P ≤N-1$, WHERE $Ω:=√(2A(N+1)(N+3))$. IN TERMS OF THIS BASIS, THE ISOTROPY SUMMANDS ARE
\BEGIN{ALIGN*}
\MATHFRAK{M}_1 &= \SPAN_{\R} \{ \XI_1\} , \QUAD
\MATHFRAK{M}_2= \SPAN_{\R} \{ \XI_2,\XI_3\} ,\QUAD
\MATHFRAK{M}_3=  \SPAN_{\R} \{E_{4P},E_{4P+1},E_{4P+2},E_{4P+3}\}_{P=1}^{N-1}.
\END{ALIGN*}
FROM (\REF{INVARIANTELEMENTSU1TWISTEDMA}) WE OBTAIN:
\BEGIN{THEOREM}\LABEL{EXPLICITSPINORSK=U1}
USING THE ABOVE ORTHONORMAL BASIS AND THE CORRESPONDING DESCRIPTION OF THE SPINOR MODULE FROM REMARK \REF{REORDEREDCLIFFALGREPRESENTATION}, THE SPACE OF INVARIANT SPINORS ON $(S^{4N-1}=\FRAC{\SP(N)\U(1)}{\SP(N-1)\U(1)}, G_{A,B,C})$ FOR ANY $A,B,C>0$ IS TRIVIAL UNLESS $N$ IS EVEN. IF $N$ IS EVEN,
\[
\SIGMA_{\INV} = \SPAN_{\C}\{ \OMEGA^{N/2}, \  Y_1\WEDGE \OMEGA^{(N-2)/2}\},
\]
WHERE $\OMEGA := \SUM_{I=1}^{N-1} Y_{2I} \WEDGE Y_{2I+1}$.
\END{THEOREM}
\BEGIN{PROOF}
THIS FOLLOWS DIRECTLY FROM (\REF{INVARIANTELEMENTSU1TWISTEDMA}). ALTERNATIVELY, ONE CAN ARGUE EXACTLY AS IN THE PROOF OF THEOREM \REF{EXAMPLE_INVARIANT_SPINORS_SP_TWISTED_7SPHERE}, USING ONLY THE SPIN LIFT OF THE OPERATOR
\BEGIN{ALIGN}\LABEL{U1EXTRAISOTROPYOPERATOR}
\AD( IF_{1,1}^{(N)},I)\RVERT_{\MATHFRAK{M}} = \XI_2\WEDGE \XI_3-\PHI_1 .
\END{ALIGN}
\END{PROOF}
IN ORDER TO DIFFERENTIATE THESE SPINORS, WE CALCULATE:
\BEGIN{LEMMA}
THE NOMIZU MAP FOR THE LEVI-CIVITA CONNECTION OF $G_{A,B,C}$ IS GIVEN BY
\BEGIN{ALIGN*}
\UPLAMBDA^{G_{A,B,C}}(X_1)X_2 &=\FRAC{1}{2}[X_1,X_2]_\MATHFRAK{M},\QUAD
\UPLAMBDA^{G_{A,B,C}}(X)Y =(1-\FRAC{A}{2B})[X,Y]_\MATHFRAK{M},\QUAD
\UPLAMBDA^{G_{A,B,C}}(Y)X = \FRAC{A}{2B} [Y,X]_\MATHFRAK{M},\\
\UPLAMBDA^{G_{A,B,C}}(Y_1)Y_2&= \FRAC{1}{2}[Y_1,Y_2]_\MATHFRAK{M},\QUAD
\UPLAMBDA^{G_{A,B,C}}(X)Z = (1-\FRAC{A}{2C})[X,Z]_\MATHFRAK{M},\QUAD
\UPLAMBDA^{G_{A,B,C}}(Z)X = \FRAC{A}{2C} [Z,X]_\MATHFRAK{M},\\
\UPLAMBDA^{G_{A,B,C}}(Z_1)Z_2&=\FRAC{1}{2}[Z_1,Z_2]_\MATHFRAK{M},\QUAD
\UPLAMBDA^{G_{A,B,C}}(Y)Z = (1-\FRAC{B}{2C}) [Y,Z]_\MATHFRAK{M},\QUAD
\UPLAMBDA^{G_{A,B,C}}(Z)Y = \FRAC{B}{2C}[Z,Y]_\MATHFRAK{M},
\END{ALIGN*}
FOR ALL $X,X_1,X_2\IN\MATHFRAK{M}_1$, $Y,Y_1,Y_2\IN\MATHFRAK{M}_2$, $Z,Z_1,Z_2\IN\MATHFRAK{M}_3$.
\END{LEMMA}
\BEGIN{PROOF}
THIS IS A STRAIGHTFORWARD CALCULATION USING THE FACT THAT $\KAPPA$ IS $\AD$-INVARIANT AND THE COMMUTATOR RELATIONS
\BEGIN{ALIGN*}
[\MATHFRAK{M}_1,\MATHFRAK{M}_1]&=0,\QUAD  [\MATHFRAK{M}_1,\MATHFRAK{M}_2]\SUBSETEQ \MATHFRAK{M}_2, \QUAD  [\MATHFRAK{M}_1,\MATHFRAK{M}_3]\SUBSETEQ \MATHFRAK{M}_3, \QUAD  [\MATHFRAK{M}_2,\MATHFRAK{M}_2]\SUBSETEQ \MATHFRAK{M}_1\OPLUS (\MATHFRAK{SP}(N-1)\OPLUS\MATHFRAK{U}(1)), \\
[\MATHFRAK{M}_2,\MATHFRAK{M}_3]&\SUBSETEQ \MATHFRAK{M}_3,\QUAD  [\MATHFRAK{M}_3,\MATHFRAK{M}_3]\SUBSETEQ \MATHFRAK{M}_1\OPLUS\MATHFRAK{M}_2\OPLUS (\MATHFRAK{SP}(N-1)\OPLUS\MATHFRAK{U}(1)) .
\END{ALIGN*}
\END{PROOF}
\BEGIN{REMARK} \LABEL{SQUASHEDUNITARYNOMIZUMAP}
IN TERMS OF OUR CHOSEN BASIS, THE OPERATORS IN THE PRECEDING LEMMA TAKE THE FORM
\BEGIN{ALIGN*}
\UPLAMBDA^{G_{A,B,C}}(\XI_1) &= \FRAC{2(1-\FRAC{A}{2B})}{\OMEGA} \  \XI_2\WEDGE \XI_3 + \FRAC{(1-\FRAC{A}{2C})}{\OMEGA} \SUM_{P=1}^{N-1}(E_{4P}\WEDGE E_{4P+1} + E_{4P+2}\WEDGE E_{4P+3}  ),\\
\UPLAMBDA^{G_{A,B,C}}(\XI_2)&= \FRAC{A}{B \OMEGA }\  \XI_3\WEDGE \XI_1 + \FRAC{(1-\FRAC{B}{2C})}{2\SQRT{B(N+1)}} \SUM_{P=1}^{N-1} (E_{4P}\WEDGE E_{4P+2} - E_{4P+1}\WEDGE E_{4P+3}   ), \\
\UPLAMBDA^{G_{A,B,C}}(\XI_3)&= \FRAC{A}{B\OMEGA }\  \XI_1\WEDGE \XI_2 + \FRAC{(1-\FRAC{B}{2C})}{2\SQRT{B(N+1)}} \SUM_{P=1}^{N-1} (E_{4P}\WEDGE E_{4P+3} + E_{4P+1}\WEDGE E_{4P+2}   ) ,
\END{ALIGN*}
IN THE VERTICAL DIRECTIONS AND, IN THE HORIZONTAL DIRECTIONS,
\BEGIN{ALIGN*}
\UPLAMBDA^{G_{A,B,C}}(E_{4P})&= -\FRAC{A}{2C\OMEGA } \ \XI_1\WEDGE E_{4P+1} - \FRAC{\SQRT{B}}{4C\SQRT{N+1}}( \XI_2\WEDGE E_{4P+2} +    \XI_3\WEDGE E_{4P+3}),\\
\UPLAMBDA^{G_{A,B,C}}(E_{4P+1})&= \FRAC{A}{2C\OMEGA } \ \XI_1\WEDGE E_{4P} + \FRAC{\SQRT{B}}{4C\SQRT{N+1}}( \XI_2\WEDGE E_{4P+3} -\XI_3\WEDGE E_{4P+2}   ), \\
\UPLAMBDA^{G_{A,B,C}}(E_{4P+2}) &= -\FRAC{A}{2C\OMEGA } \ \XI_1\WEDGE E_{4P+3} + \FRAC{\SQRT{B}}{4C\SQRT{N+1}}(  \XI_2\WEDGE E_{4P} +\XI_3\WEDGE E_{4P+1}  ) ,\\
\UPLAMBDA^{G_{A,B,C}}(E_{4P+3}) &= \FRAC{A}{2C\OMEGA } \ \XI_1\WEDGE E_{4P+2} - \FRAC{\SQRT{B}}{4C\SQRT{N+1}} ( \XI_2\WEDGE E_{4P+1} - \XI_3\WEDGE E_{4P}   ).
\END{ALIGN*}
\END{REMARK}
IN ODD DIMENSIONS, ALMOST CONTACT STRUCTURES ARE NATURAL CANDIDATES FOR INTERESTING GEOMETRIES; IN THE FOLLOWING PROPOSITION WE STUDY THE EXISTENCE OF INVARIANT ALMOST CONTACT STRUCTURES FOR THESE SPHERES.
\BEGIN{PROPOSITION} \LABEL{U1TWISTEDSASAKIAN}
THE SPHERE $(S^{4N-1}=\FRAC{\SP(N) \U(1)}{\SP(N-1) \U(1)}, G_{A,B,C})$ ADMITS:
\BEGIN{ENUMERATE}[(I)] \ITEM A COMPATIBLE INVARIANT NORMAL ALMOST CONTACT METRIC STRUCTURE FOR ALL $A,B,C >0$.
\ITEM A COMPATIBLE INVARIANT $\ALPHA$-CONTACT STRUCTURE IF AND ONLY IF $\FRAC{A}{B\OMEGA} = \FRAC{A}{2C\OMEGA }=\ALPHA$.
\ITEM A COMPATIBLE INVARIANT $\ALPHA$-K-CONTACT STRUCTURE IF AND ONLY IF $\FRAC{A}{B\OMEGA} = \FRAC{A}{2C\OMEGA }=\ALPHA$.
\END{ENUMERATE}
IN PARTICULAR THERE EXISTS A COMPATIBLE INVARIANT $\ALPHA$-SASAKIAN STRUCTURE IF AND ONLY IF $\FRAC{A}{B\OMEGA} = \FRAC{A}{2C\OMEGA }=\ALPHA$.
\END{PROPOSITION}
\BEGIN{PROOF}
FOLLOWING A SIMILAR ARGUMENT AS IN THE PROOF OF PROPOSITION \REF{SPECIALUNITARYALPHASASAKIAN}, THE ONLY CHOICES FOR THE REEB VECTOR FIELD ARE $\XI:= \PM \XI_1$. WE CLAIM THAT THE 2-FORM $\PHI:=G_{A,B,C}(\CDOT, \VARPHI(\CDOT))$ IS INVARIANT IF AND ONLY IF
\BEGIN{ALIGN} \LABEL{INVARIANT2FORMSPACE}
\PHI \IN (\LAMBDA^2 \MATHFRAK{M}_2)^{\SP(N-1) \U(1)} \OPLUS (\LAMBDA^2\MATHFRAK{M}_3)^{\SP(N-1) \U(1)}\SIMEQ \SPAN_{\R} \{  \AD\XI\RVERT_{\MATHFRAK{M}_2}  \} \OPLUS \SPAN_{\R} \{  \AD\XI\RVERT_{\MATHFRAK{M}_3}  \} .
\END{ALIGN}
THE PROOF OF THE CLAIM CONSISTS OF TWO PARTS: FIRST, BY NOTING THAT $\MATHFRAK{M}_1$, $\MATHFRAK{M}_2, \MATHFRAK{M}_3$ ARE IRREDUCIBLE AND HAVE PAIRWISE DISTINCT DIMENSIONS, WE SEE THAT ANY INVARIANT 2-FORM HAS TRIVIAL $\MATHFRAK{M}_1\OTIMES \MATHFRAK{M}_2$, $\MATHFRAK{M}_1\OTIMES \MATHFRAK{M}_3$, AND $\MATHFRAK{M}_2\OTIMES \MATHFRAK{M}_3$ COMPONENTS. SECOND, WE NOTE THAT $\LAMBDA^2\MATHFRAK{M}_1=0$ (FOR DIMENSION REASONS) AND THAT EACH $\LAMBDA^2 \MATHFRAK{M}_I$ $(I=2,3)$ HAS A REAL 3-DIMENSIONAL SPACE OF $\SP(N-1)$-INVARIANT 2-FORMS, CORRESPONDING TO THE QUATERNIONIC STRUCTURE
\[
\MATHCAL{I}_I:= \PHI_1\RVERT_{\MATHFRAK{M}_I}, \QUAD \MATHCAL{J}_I:=  \PHI_2\RVERT_{\MATHFRAK{M}_I},\QUAD \MATHCAL{K}_I:= \PHI_3\RVERT_{\MATHFRAK{M}_I}.
\]
IMPOSING THE ADDITIONAL CONDITION OF $\U(1)$-INVARIANCE IS EQUIVALENT TO ALSO REQUIRING $\MATHCAL{I}$-COMPLEX LINEARITY, GIVING $(\LAMBDA^2\MATHFRAK{M}_I)^{\SP(N-1)\U(1)} \SIMEQ \SPAN_{\R}\{\MATHCAL{I}_I\} = \SPAN_{\R} \{  \AD\XI\RVERT_{\MATHFRAK{M}_I}  \}  $ FOR $I=2,3$.

RETURNING TO THE MAIN PROOF, (\REF{INVARIANT2FORMSPACE}) IS EQUIVALENT TO $\VARPHI = \LAMBDA_1 \AD\XI\RVERT_{\MATHFRAK{M}_2} \OPLUS \LAMBDA_2 \AD\XI\RVERT_{\MATHFRAK{M}_3} $ FOR SOME $\LAMBDA_1,\LAMBDA_2 \IN\R$, AND THE METRIC COMPATIBILITY CONDITION $G_{A,B,C}(\VARPHI(X),\VARPHI(Y)) = G_{A,B,C}(X,Y)- G_{A,B,C}(\XI,X)G_{A,B,C}(\XI,Y)$ NECESSITATES $\LAMBDA_1 = \OMEGA/2$, $\LAMBDA_2=\OMEGA $, I.E.
\[
\VARPHI =\FRAC{\OMEGA }{2} \AD\XI\RVERT_{\MATHFRAK{M}_2} \OPLUS \OMEGA \AD\XI\RVERT_{\MATHFRAK{M}_3}.
\]
%USING THE COMMUTATOR RELATIONS
%[\XI_1,\XI_2]_{\MATHFRAK{M}} &= \FRAC{2}{\OMEGA} \XI_3, \QUAD   [\XI_1,\XI_3]_{\MATHFRAK{M}}=-\FRAC{2}{\OMEGA} \XI_2, \QUAD  [\XI_1,E_{4P}]_{\MATHFRAK{M}} = \FRAC{1}{\OMEGA}  E_{4P+1}, \\
%[\XI_1,E_{4P+1}]_{\MATHFRAK{M}}&= -\FRAC{1}{\OMEGA}  E_{4P} , \QUAD  [\XI_1,E_{4P+2}]_{\MATHFRAK{M}}= \FRAC{1}{\OMEGA} E_{4P+3},\QUAD [\XI_1,E_{4P+3}]_{\MATHFRAK{M}}= -\FRAC{1}{\OMEGA} E_{4P+2}, \\
%[\XI_2,\XI_3]_{\MATHFRAK{M}}&= \FRAC{2A}{B\OMEGA }\XI_1 , \QUAD  [\XI_2,E_{4P}]_{\MATHFRAK{M}}= \FRAC{1}{2\SQRT{B(N+1)}}E_{4P+2}, \QUAD  [\XI_2,E_{4P+1}]_{\MATHFRAK{M}} = \FRAC{-1}{2\SQRT{B(N+1)}} E_{4P+3}, \\
%[\XI_2,E_{4P+2} ]_{\MATHFRAK{M}} &= \FRAC{-1}{2\SQRT{B(N+1)}} E_{4P} , \QUAD  [\XI_2,E_{4P+3}]_{\MATHFRAK{M}} = \FRAC{1}{2\SQRT{B(N+1)}} E_{4P+1}, \\
%[\XI_3,E_{4P}]_{\MATHFRAK{M}} &= \FRAC{1}{2\SQRT{B(N+1)}} E_{4P+3} , \QUAD  [\XI_3,E_{4P+1}]_{\MATHFRAK{M}} = \FRAC{1}{2\SQRT{B(N+1)}} E_{4P+2}, \\
%[\XI_3,E_{4P+2}]_{\MATHFRAK{M}} &= \FRAC{-1}{2\SQRT{B(N+1)}} E_{4P+1} , \QUAD  [\XI_3,E_{4P+3}]_{\MATHFRAK{M}}= \FRAC{-1}{2\SQRT{B(N+1)}} E_{4P},
%[E_{4P},E_{4Q}]_{\MATHFRAK{M}}&= [E_{4P+1},E_{4Q+1}]_{\MATHFRAK{M}} = [E_{4P+2},E_{4Q+2}]_{\MATHFRAK{M}} = [E_{4P+3},E_{4Q+3}]_{\MATHFRAK{M}}=0, \\
%[E_{4P},E_{4Q+1}]_{\MATHFRAK{M}}&= \FRAC{A \DELTA_{PQ} }{C\OMEGA } \XI_1, \QUAD  [E_{4P},E_{4Q+2}]_{\MATHFRAK{M}}= \FRAC{\SQRT{B}\DELTA_{PQ}}{2C\SQRT{N+1}}\XI_2 ,\QUAD  [E_{4P},E_{4Q+3}]_{\MATHFRAK{M}}= \FRAC{\SQRT{B}\DELTA_{PQ}}{2C\SQRT{N+1}} \XI_3 ,  \\
%[E_{4P+1},E_{4Q+2}]_{\MATHFRAK{M}} &= \FRAC{\SQRT{B}\DELTA_{PQ}}{2C\SQRT{N+1}}\XI_3, \QUAD  [E_{4P+1},E_{4Q+3}]_{\MATHFRAK{M}} = \FRAC{-\SQRT{B}\DELTA_{PQ}}{2C\SQRT{N+1}}\XI_2, \QUAD  [E_{4P+2},E_{4Q+3}]_{\MATHFRAK{M}}= \FRAC{A\DELTA_{PQ}}{C\OMEGA }\XI_1,
ONE THEN CALCULATES THAT THE NIJENHUIS TENSOR VANISHES FOR ANY VALUES OF $A,B,C$, AND THE STRUCTURE IS $\ALPHA$-CONTACT ($D\ETA = 2\ALPHA \PHI$) AND $\ALPHA$-K-CONTACT ($\NABLA^G_X\XI = -\ALPHA\VARPHI(X)$) IF AND ONLY IF $\FRAC{A}{B\OMEGA}= \FRAC{A}{2C\OMEGA }=\ALPHA$.
\END{PROOF}
SOLVING THE EQUATIONS IN THE PRECEDING PROPOSITION FOR $A,B,C$, WE IMMEDIATELY OBTAIN:
\BEGIN{COROLLARY}\LABEL{SPNU1ALPHASASAKIANPARAMTERIZATION}
FOR EACH FIXED $\ALPHA>0$, THE INVARIANT $\ALPHA$-SASAKIAN STRUCTURES OCCUR IN A 1-PARAMETER FAMILY:
\BEGIN{ALIGN}
A=8\LAMBDA^2\ALPHA^2(N+1)(N+3),\QUAD B=2\LAMBDA, \QUAD C=\LAMBDA, \QQUAD (\LAMBDA>0). \LABEL{INVSASAKIANFAMILY}
\END{ALIGN}
IN PARTICULAR, THE ROUND METRIC OCCURS FOR THE PARAMETER $\LAMBDA=\FRAC{1}{8(N+1)}$.
\END{COROLLARY}
IN CONTRAST TO THE CASE $G=(N)$, WE SHALL SEE IN THE FOLLOWING REMARK THAT THE RELATIONSHIP BETWEEN INVARIANT EINSTEIN-SASAKIAN STRUCTURES AND INVARIANT KILLING SPINORS IS MORE COMPLICATED FOR $G=(N)(1)$, STEMMING FROM THE FACT THAT THIS GROUP HAS A NON-TRIVIAL $2$-DIMENSIONAL COMPLEX REPRESENTATION.
\BEGIN{REMARK}
WE REMARK THAT THE ABOVE 2-FORM $\PHI$ TAKES THE FORM (\REF{PHI1}). THEREFORE IF AN INVARIANT EINSTEIN-SASAKIAN STRUCTURE EXISTS THEN PROPOSITION \REF{E1SPINORBASIS} AND THEOREM 1 IN [24] IMPLY THAT THERE ARE TWO LINEARLY INDEPENDENT KILLING SPINORS INSIDE $E_1^- = \SPAN_{\C}\{1,\ Y_1\WEDGE \OMEGA^{N-1}\}$. IF $N=1$ THEN $\DIM_{\C} \SIGMA =2$, AND IT FOLLOWS THAT $\SIGMA=E_1^-$ IS SPANNED BY TWO LINEARLY INDEPENDENT (NON-INVARIANT) KILLING SPINORS. IN FACT, FOR ALL $N\GEQ 1$, ONE SEES FROM THEOREM \REF{EXPLICITSPINORSK=U1} THAT THE INTERSECTION OF $E_1^-$ WITH THE SPACE OF INVARIANT SPINORS IS TRIVIAL, SO THE KILLING SPINORS SPANNING $E_1^-$ ARE NOT INVARIANT. THIS CONTRASTS WITH THE BEHAVIOUR IN THE CASES $G=\SP(N)$, $\SU(N+1)$, WHERE THE SPACES $E_I^{\PM}$ HAD BASES OF INVARIANT SPINORS. INDEED, IN EACH OF THE THREE CASES $G=\SP(N)\U(1)$, $\SP(N)$, $\SU(N+1)$ IT IS EASY TO SEE BY ARGUING SIMILARLY AS IN THE PROOF OF \CITE[PROP.\@ 7.1]{KATH_TDUALS} THAT THERE IS A $G$-REPRESENTATION ON $E_{I}^{\PM}$, AND IN THE LATTER TWO CASES WE USE THE FACT THAT $\DIM_{\C} E_I^{\PM}\LEQ 2$ TO CONCLUDE THAT THE REPRESENTATION MUST BE TRIVIAL (EQUIVALENTLY, THE KILLING SPINORS SPANNING $E_I^{\PM}$ ARE INVARIANT). HOWEVER, UNLIKE THE OTHER TWO GROUPS, $G=\SP(N) \U(1)$ HAS A NONTRIVIAL 2-DIMENSIONAL REPRESENTATION, WHICH IS WHAT ALLOWS THE KILLING SPINORS SPANNING $E_1^-$ TO BE NON-INVARIANT IN THIS CASE. THIS BEHAVIOUR IS SOMEWHAT SURPRISING; ONE WOULD INTUITIVELY EXPECT THE KILLING SPINORS ASSOCIATED TO AN INVARIANT EINSTEIN-SASAKIAN STRUCTURE TO ALSO BE INVARIANT.
\END{REMARK}
\BEGIN{THEOREM} \LABEL{U1TWISTEDGKS}
THE SPHERE $(S^{4N-1}=\FRAC{\SP(N) \U(1)}{\SP(N-1) \U(1)}, G_{A,B,C})$ ADMITS AN INVARIANT GENERALIZED KILLING SPINOR IF AND ONLY IF $N=2$. IF $N=2$ THERE EXISTS A PAIR $\PSI_0$, $\PSI_1$ OF LINEARLY INDEPENDENT INVARIANT GENERALIZED KILLING SPINORS,
\[
\NABLA_X^{G_{A,B,C}} \PSI_I = A_I(X)\CDOT \PSI_I , \QUAD I=0,1,
\]
FOR THE ENDOMORPHISMS
\BEGIN{ALIGN*}
A_0&:=\FRAC{A}{2\OMEGA }(\FRAC{1}{C}-\FRAC{1}{B}) \ID \RVERT_{\MATHFRAK{M}_1} + \LEFT(\FRAC{A}{2B\OMEGA }- \FRAC{(1-\FRAC{B}{2C})}{2\SQRT{3B}}\RIGHT)\ID\RVERT_{\MATHFRAK{M}_2} +  \LEFT(  -\FRAC{A}{4C\OMEGA } -\FRAC{\SQRT{B}}{4C\SQRT{3}} \RIGHT)\ID\RVERT_{\MATHFRAK{M}_3}  ,  \\
A_1&:= \FRAC{A}{2\OMEGA }(\FRAC{1}{C}-\FRAC{1}{B}) \ID \RVERT_{\MATHFRAK{M}_1} + \LEFT(\FRAC{A}{2B\OMEGA }+ \FRAC{(1-\FRAC{B}{2C})}{2\SQRT{3B}}\RIGHT)\ID\RVERT_{\MATHFRAK{M}_2} +  \LEFT(  -\FRAC{A}{4C\OMEGA } +\FRAC{\SQRT{B}}{4C\SQRT{3}} \RIGHT)\ID\RVERT_{\MATHFRAK{M}_3}
\END{ALIGN*}
WITH AT MOST THREE DISTINCT EIGENVALUES.
\END{THEOREM}
\BEGIN{PROOF}
SUPPOSE THERE EXISTS AN INVARIANT GENERALIZED KILLING SPINOR $\NABLA_X^{G_{A,B,C}}\PSI=A(X)\CDOT \PSI$ AND, USING THEOREM \REF{EXPLICITSPINORSK=U1}, WRITE $\PSI = \LAMBDA_1\OMEGA^{\FRAC{N}{2}} + \LAMBDA_2 Y_1\WEDGE \OMEGA^{\FRAC{N-2}{2}}$ FOR SOME $\LAMBDA_1,\LAMBDA_2\IN\C$. LET US EXAMINE THE SPIN LIFT OF THE OPERATOR $\UPLAMBDA^{G_{A,B,C}}(E_{4P})$. ONE CALCULATES
\BEGIN{ALIGN*}
\FRAC{1}{2}\XI_1\CDOT E_{4P+1} \CDOT \PSI  &= \FRAC{I}{2}\LEFT[ \LAMBDA_1\LEFT( \FRAC{N}{2}Y_{2P+1}\WEDGE \OMEGA^{\FRAC{N-2}{2}} -Y_{2P}\WEDGE \OMEGA^{\FRAC{N}{2}} \RIGHT) + \LAMBDA_2\LEFT(\FRAC{N-2}{2}Y_1\WEDGE Y_{2P+1}\WEDGE \OMEGA^{\FRAC{N-4}{2}} -Y_1\WEDGE Y_{2P} \WEDGE \OMEGA^{\FRAC{N-2}{2}}    \RIGHT)     \RIGHT] , \\
\FRAC{1}{2}\XI_2\CDOT E_{4P+2}\CDOT \PSI &= \FRAC{1}{2}\LEFT[ \LAMBDA_1\LEFT( \FRAC{N}{2}Y_1\WEDGE Y_{2P}\WEDGE \OMEGA^{\FRAC{N-2}{2}} - Y_1\WEDGE Y_{2P+1} \WEDGE \OMEGA^{\FRAC{N}{2}}   \RIGHT) + \LAMBDA_2\LEFT(  Y_{2P+1}\WEDGE \OMEGA^{\FRAC{N-2}{2}} - \FRAC{N-2}{2} Y_{2P} \WEDGE \OMEGA^{\FRAC{N-4}{2}} \RIGHT)   \RIGHT] ,  \\
\FRAC{1}{2}\XI_3\CDOT E_{4P+3}\CDOT \PSI &= \FRAC{1}{2}\LEFT[ \LAMBDA_1\LEFT( \FRAC{N}{2}Y_1\WEDGE Y_{2P} \WEDGE \OMEGA^{\FRAC{N-2}{2}}+Y_1\WEDGE Y_{2P+1}\WEDGE \OMEGA^{\FRAC{N}{2}}\RIGHT)+ \LAMBDA_2\LEFT( Y_{2P+1}\WEDGE \OMEGA^{\FRAC{N-2}{2}} +\FRAC{N-2}{2}Y_{2P}\WEDGE \OMEGA^{\FRAC{N-4}{2}}\RIGHT)     \RIGHT] ,
\END{ALIGN*}
AND THE EXPLICIT FORMULA FROM REMARK \REF{SQUASHEDUNITARYNOMIZUMAP} THEN GIVES
\BEGIN{ALIGN*}
\WIDETILDE{\UPLAMBDA^{G_{A,B,C}}}(E_{4P})\CDOT \PSI &=  -\LEFT( \FRAC{IAN\LAMBDA_1}{8C\OMEGA } + \FRAC{\SQRT{B}\LAMBDA_2}{4C\SQRT{N+1}} \RIGHT) Y_{2P+1}\WEDGE \OMEGA^{\FRAC{N-2}{2}} +   \LEFT( \FRAC{IA\LAMBDA_2}{4C\OMEGA } + \FRAC{N\LAMBDA_1}{2} \RIGHT) Y_1\WEDGE Y_{2P} \WEDGE \OMEGA^{\FRAC{N-2}{2}} \\
&\QQUAD + \LEFT( \FRAC{IA\LAMBDA_1}{4C\OMEGA }\RIGHT) Y_{2P}\WEDGE \OMEGA^{\FRAC{N}{2}} - \LEFT( \FRAC{IA(N-2)\LAMBDA_2}{8C\OMEGA } \RIGHT) Y_1\WEDGE Y_{2P+1}\WEDGE \OMEGA^{\FRAC{N-4}{2}}
\END{ALIGN*}
IN ORDER FOR THE RIGHT HAND SIDE OF THIS EXPRESSION TO BE EQUAL TO THE CLIFFORD PRODUCT OF A (REAL) TANGENT VECTOR WITH $\PSI$, ONE SEES THAT SUCH A VECTOR MUST BE OF THE FORM $SE_{4P} + TE_{4P+1}$ FOR SOME $S,T\IN \R$. COMPARING WITH
\BEGIN{ALIGN*}
(SE_{4P} +TE_{4P+1})\CDOT \PSI &= \LEFT( \FRAC{INS\LAMBDA_1}{2} - \FRAC{NT\LAMBDA_1}{2}  \RIGHT) Y_{2P+1}\WEDGE \OMEGA^{\FRAC{N-2}{2}} +(IS\LAMBDA_1 +T\LAMBDA_1)Y_{2P}\WEDGE \OMEGA^{\FRAC{N}{2}}\\
&\QQUAD  + \LEFT(-\FRAC{I(N-2)S\LAMBDA_2}{2} +\FRAC{(N-2)T\LAMBDA_2 }{2}\RIGHT) Y_1\WEDGE Y_{2P+1}\WEDGE \OMEGA^{\FRAC{N-4}{2}} \\
&\QQUAD +  (-IS\LAMBDA_2- T\LAMBDA_2)Y_1\WEDGE Y_{2P}\WEDGE \OMEGA^{\FRAC{N-2}{2}}
\END{ALIGN*} GIVES THE NECESSARY CONDITIONS
\BEGIN{ALIGN*}
\LEFT( \FRAC{IA\LAMBDA_1}{4C\OMEGA }\RIGHT) Y_{2P}\WEDGE \OMEGA^{\FRAC{N}{2}} &= (IS\LAMBDA_1 +T\LAMBDA_1)Y_{2P}\WEDGE \OMEGA^{\FRAC{N}{2}} , \\
\LEFT( \FRAC{IA\LAMBDA_2}{4C\OMEGA } + \FRAC{N\LAMBDA_1}{2} \RIGHT) Y_1\WEDGE Y_{2P} \WEDGE \OMEGA^{\FRAC{N-2}{2}} &= -(IS\LAMBDA_1 +T\LAMBDA_1) Y_1\WEDGE Y_{2P}\WEDGE\OMEGA^{\FRAC{N-2}{2}} .
\END{ALIGN*}
IF $N\NEQ 2$ THEN $Y_{2P}\WEDGE \OMEGA^{\FRAC{N}{2}}$ AND $Y_1\WEDGE Y_{2P}\WEDGE \OMEGA^{\FRAC{N-2}{2}} $ ARE NON-ZERO (THE LATTER IS NON-ZERO INDEPENDENT OF $N$), AND WE HAVE
\[
\LEFT( \FRAC{IA\LAMBDA_1}{4C\OMEGA }\RIGHT) = -\LEFT( \FRAC{IA\LAMBDA_2}{4C\OMEGA } + \FRAC{N\LAMBDA_1}{2} \RIGHT) .
\]
IT FOLLOWS BY TAKING THE REAL PART OF BOTH SIDES THAT $\LAMBDA_1=\LAMBDA_2=0$, I.E. $\PSI\EQUIV 0$ IS TRIVIAL.
FOR $N=2$, LIFTING THE OPERATORS IN REMARK \REF{SQUASHEDUNITARYNOMIZUMAP} AND APPLYING THEM TO THE SPINORS
\BEGIN{ALIGN*}
\PSI_0&:= \FRAC{1}{\SQRT{2}}(\OMEGA +IY_1),\QUAD \PSI_1:=\XI_1\CDOT \PSI_0 = \FRAC{1}{\SQRT{2}}(I\OMEGA +Y_1)
\END{ALIGN*}
GIVES THE RESULT.
\END{PROOF}
NEXT, WE CALCULATE THE AMBROSE-SINGER TORSION AND DETERMINE ITS TYPE:
\BEGIN{PROPOSITION}
FOR ANY $A,B,C>0$ THE SPHERE $(S^{4N-1}=\FRAC{\SP(N)\U(1)}{\SP(N-1)\U(1)}, G_{A,B,C})$ HAS AMBROSE-SINGER TORSION OF TYPE $\MATHCAL{T}_{\TOTALLYSKEW}\OPLUS \MATHCAL{T}_{\CT}$, GIVEN BY {\SMALL
\BEGIN{ALIGN*}
T^{\AS}(\XI_1,\XI_2)&= -\FRAC{2}{\OMEGA} \XI_3 , \QUAD T^{\AS}(\XI_1,\XI_3) =  \FRAC{2}{\OMEGA} \XI_2  , \QUAD  T^{\AS}(\XI_2,\XI_3) = - \FRAC{2A}{B\OMEGA}\XI_1 ,  \\
T^{\AS}(\XI_1, - )\RVERT_{\MATHFRAK{M}_3} &= \FRAC{1}{\OMEGA}\PHI_1\RVERT_{\MATHFRAK{M}_3}, \QUAD T^{\AS}(\XI_2, - )\RVERT_{\MATHFRAK{M}_3} = \FRAC{1}{2\SQRT{B(N+1)}}\PHI_2\RVERT_{\MATHFRAK{M}_3}, \QUAD  T^{\AS}(\XI_3, - )\RVERT_{\MATHFRAK{M}_3} = \FRAC{1}{2\SQRT{B(N+1)}}\PHI_3\RVERT_{\MATHFRAK{M}_3},        \\
T^{\AS}(E_{4P},E_{4Q}) &= T^{\AS}(E_{4P+1},E_{4Q+1}) = T^{\AS}(E_{4P+2},E_{4Q+2}) = T^{\AS}(E_{4P+3},E_{4Q+3})=0\\
T^{\AS} (E_{4P},E_{4Q+1}) &= - \FRAC{A\DELTA_{P,Q}}{C\OMEGA } \XI_1 , \QUAD T^{\AS}(E_{4P},E_{4Q+2}) = -\FRAC{\SQRT{B} \DELTA_{P,Q} }{2C\SQRT{N+1}} \XI_2, \QUAD T^{\AS}(E_{4P},E_{4Q+3}) = -\FRAC{\SQRT{B} \DELTA_{P,Q} }{2C\SQRT{N+1}} \XI_3, \\
T^{\AS}(E_{4P+1},E_{4Q+2}) &= -\FRAC{\SQRT{B} \DELTA_{P,Q} }{2C\SQRT{N+1}} \XI_3,\QUAD  T^{\AS} (E_{4P+1},E_{4Q+3}) = \FRAC{\SQRT{B} \DELTA_{P,Q} }{2C\SQRT{N+1}} \XI_2, \QUAD T^{\AS}(E_{4P+2},E_{4Q+3}) = - \FRAC{A\DELTA_{P,Q}}{C\OMEGA} \XI_1,
\END{ALIGN*}
FOR $P,Q=1,\DOTS, N-1$, WHERE $\PHI_1,\PHI_2,\PHI_3$ ARE DEFINED FORMALLY AS IN (\REF{PHI1})-(\REF{PHI3}). THE PROJECTION OF $T^{\AS}$ ONTO $\MATHCAL{T}_{\TOTALLYSKEW}$ IS
\[
T^{\AS}_{\TOTALLYSKEW} = - \LEFT( \FRAC{2A+4B}{3B\OMEGA}  \RIGHT)\XI_{1,2,3}   + \LEFT(\FRAC{A+2C}{3C\OMEGA} \RIGHT) \XI_1\WEDGE \PHI_1\RVERT_{\MATHFRAK{M}_3}
+\LEFT( \FRAC{B+2C}{6C\SQRT{B(N+1)}}  \RIGHT)\SUM_{I=2}^3  \XI_I \WEDGE \PHI_I\RVERT_{\MATHFRAK{M}_3}   ,
\]
WITH $T^{\AS} = T^{\AS}_{\TOTALLYSKEW}$ IF AND ONLY IF $A=B=C$.
\END{PROPOSITION}
BEFORE STUDYING THE SITUATION IN DIMENSION $7$ IN MORE DETAIL, WE CONCLUDE OUR DISCUSSION OF THE GENERAL CASE WITH A DESCRIPTION OF THE INVARIANT DIFFERENTIAL FORMS OF DEGREE $≤3$:
\BEGIN{PROPOSITION}
THE INVARIANT DIFFERENTIAL FORMS OF DEGREE LESS THAN OR EQUAL TO $3$ ON $(S^{4N-1}=\FRAC{\SP(N)\U(1)}{\SP(N-1)\U(1)}, G_{A,B,C})$ ARE GIVEN IN TABLE \REF{TAB:INV_FORMS_SPNU1}.
\END{PROPOSITION}
\BEGIN{PROOF}
THE RESULT FOR DEGREE $K=1$ IS CLEAR, AND FOR $K=2$ IT FOLLOWS FROM THE PROOF OF PROPOSITION \REF{U1TWISTEDSASAKIAN}. FOR $K=3$, ONE VERIFIES THAT THE FORMS GIVEN IN TABLE  \REF{TAB:INV_FORMS_SPNU1} ARE PRECISELY THE $\SP(N-1)$-INVARIANT $3$-FORMS IN $\SPAN_{\C}\{\XI_{1,2,3}, \XI_I\WEDGE \PHI_J\RVERT_{\MATHFRAK{M}_3}\}_{I,J=1,2,3}$ WHICH ARE ANNIHILATED BY THE ADDITIONAL OPERATOR (\REF{U1EXTRAISOTROPYOPERATOR}).
\END{PROOF}
\BEGIN{TABLE}[H!]
\CENTERING
\CAPTION{INVARIANT FORMS OF LOW DEGREE ON $(S^{4N-1}=\FRAC{\SP(N)\U(1)}{\SP(N-1)\U(1)},G_{A,B,C})$}
\BEGIN{TABULAR}{|L||L|L|}\HLINE
$K$  & $\DIM \LAMBDA^K_{\INV}$ & BASIS FOR $ \LAMBDA^K_{\INV}$ \\\HLINE
$0$    & $1$ & $1$  \\
$1$ & $1$ & $\XI_1 $  \\
$2$ & $2$ & $\PHI_1\RVERT_{\MATHFRAK{M}_2}$, $\PHI_1\RVERT_{\MATHFRAK{M}_3}$ \\
$3$   & $4$ &  $\XI_{1,2,3}$, $\XI_1\WEDGE \PHI_1\RVERT_{\MATHFRAK{M}_3}$, $(\XI_2\WEDGE \PHI_2\RVERT_{\MATHFRAK{M}_3}+\XI_3\WEDGE \PHI_3\RVERT_{\MATHFRAK{M}_3})$, $(\XI_2\WEDGE \PHI_3\RVERT_{\MATHFRAK{M}_3}-\XI_3\WEDGE \PHI_2\RVERT_{\MATHFRAK{M}_3})$  \\ \HLINE
\END{TABULAR}
\LABEL{TAB:INV_FORMS_SPNU1}
\END{TABLE}
IN WHAT FOLLOWS, WE EXAMINE MORE CLOSELY THE GENERALIZED KILLING SPINORS IN DIMENSION $7$ AND COMPARE WITH THE KNOWN RESULTS FOR THE ROUND ($3$-SASAKIAN) METRIC AND THE SECOND EINSTEIN METRIC (SEE [25, 5, 40]).
\BEGIN{REMARK}\LABEL{U1TWISTEDROUNDMETRIC}
IN THE CASE OF AN INVARIANT SASAKIAN METRIC, SUBSTITUTING (\REF{INVSASAKIANFAMILY}) INTO THE ENDOMORPHISMS FROM THE PRECEDING THEOREM GIVES
\BEGIN{ALIGN*}
A_0 &= \FRAC{1}{2} \ID\RVERT_{\MATHFRAK{M}_1\OPLUS \MATHFRAK{M}_2} + \FRAC{-\SQRT{6\LAMBDA}-1}{2\SQRT{6\LAMBDA}} \ID\RVERT_{\MATHFRAK{M}_3},  \QQUAD	A_1= \FRAC{1}{2} \ID\RVERT_{\MATHFRAK{M}_1\OPLUS \MATHFRAK{M}_2} + \FRAC{-\SQRT{6\LAMBDA}+1}{2\SQRT{6\LAMBDA}} \ID\RVERT_{\MATHFRAK{M}_3} .
\END{ALIGN*}
IT IS CLEAR THAT $A_0$ IS NEVER A MULTIPLE OF THE IDENTITY MAP, AND $A_1$ IS A MULTIPLE OF THE IDENTITY IF AND ONLY IF $\LAMBDA=1/{24}$, WHICH OCCURS PRECISELY WHEN $G_{A,B,C}$ IS THE ROUND METRIC (SEE COROLLARY \REF{SPNU1ALPHASASAKIANPARAMTERIZATION}). WHEN $\LAMBDA=1/{24}$, WE HAVE
\BEGIN{ALIGN*} A_0&=\FRAC{1}{2}\ID\RVERT_{\MATHFRAK{M}_1\OPLUS\MATHFRAK{M}_2} -\FRAC{3}{2}\ID\RVERT_{\MATHFRAK{M}_3}  ,  \QQUAD 	A_1= \FRAC{1}{2}\ID.
\END{ALIGN*}
IN THIS CASE THERE ARE THREE LINEARLY INDEPENDENT KILLING SPINORS (THE INVARIANT KILLING SPINOR $\PSI_1$ AND TWO NON-INVARIANT LINEARLY INDEPENDENT KILLING SPINORS INSIDE $E_1^-$), AND THEOREM 6 IN [24] THEN IMPLIES THE EXISTENCE OF A $3$-SASAKIAN STRUCTURE, AS EXPECTED. STARTING WITH THE GENERALIZED KILLING SPINOR $\PSI_0$, THE EXISTENCE OF A $3$-SASAKIAN STRUCTURE ALSO FOLLOWS FROM \CITE[THM.\@ 4.10]{GKSSPHERES}, WHICH NOTES MOREOVER THAT $\PSI_0$ CORRESPONDS TO THE CANONICAL SPINOR DESCRIBED IN [5] FOR A GENERAL 3-SASAKIAN MANIFOLD OF DIMENSION $7$ (AND WE THEN SEE THAT $\PSI_1=\XI_I\CDOT \PSI_0$ CORRESPONDS TO ONE OF THE AUXILIARY SPINORS DESCRIBED THEREIN).
\END{REMARK}
FINALLY, AS IN REMARK \REF{SPN_SECOND_EINSTEIN_METRIC}, WE COMMENT ON THE SECOND EINSTEIN METRIC IN THIS EXAMPLE:
\BEGIN{REMARK}
HERE, THE SECOND EINSTEIN METRIC IS GIVEN BY $G_{A,B,C}\RVERT_{A=\FRAC{1}{24}, B=\FRAC{1}{60} , C= \FRAC{1}{24}}$, AND IT IS CLEAR THAT THIS METRIC IS NOT PART OF THE FAMILY (\REF{INVSASAKIANFAMILY}). SUBSTITUTING THESE VALUES OF $A,B,C$ INTO THE ENDOMORPHISMS FROM THEOREM \REF{U1TWISTEDGKS} GIVES
\[
A_0 = -\FRAC{3}{2\SQRT{5}} \ID, \QQUAD A_1= -\FRAC{3}{2\SQRT{5}} \ID\RVERT_{\MATHFRAK{M}_1} + \FRAC{13}{2\SQRT{5}} \ID\RVERT_{\MATHFRAK{M}_2} + \FRAC{1}{2\SQRT{5}}\ID\RVERT_{\MATHFRAK{M}_3},
\]
SO $\PSI_0$ IS THE KILLING SPINOR DETERMINING THE PROPER NEARLY PARALLEL $\G_2$-STRUCTURE AND $\PSI_1$ IS A GENERALIZED KILLING SPINOR WITH $3$ DISTINCT EIGENVALUES.
\END{REMARK}

\SECTION{EXCEPTIONAL SPHERES}
\SUBSECTION{$S^6=\G_2/\SU(3)$}
THE ISOTROPY REPRESENTATION HERE IS IRREDUCIBLE, SO THE ONLY INVARIANT METRICS ARE OBTAINED FROM MULTIPLES OF THE KILLING FORM. FOR CONVENIENCE WE CHOOSE THE INVARIANT INNER PRODUCT $G=B_0$ ON $𝔤_2$, AND CONSIDER THE REDUCTIVE COMPLEMENT $𝔪:= (𝔰𝔲(3))^⊥$ WITH RESPECT TO THIS INNER PRODUCT. FOLLOWING THE NOTATION OF PROPOSITION \REF{ONBASISG2SU3}, WE HAVE
\BEGIN{ALIGN*}
\MATHFRAK{G}_2&= \SPAN_{\R} \{ \NU_1,\DOTS, \NU_{14}   \}, \QUAD
\MATHFRAK{SU}(3)= \SPAN_{\R} \{  \NU_1,\DOTS \NU_8 \},
\END{ALIGN*}
AND FOR $𝔪$ WE TAKE THE $G$-ORTHONORMAL BASIS GIVEN BY
\BEGIN{ALIGN*}
\MATHFRAK{M}=\SPAN_{\R}\{ E_1,\DOTS ,E_6\}, \ \ \TEXT{WHERE } E_I:= \BEGIN{CASES} \NU_{8+I} & I= 1,2,4,6, \\ -\NU_{8+I} & I = 3,5   .   \END{CASES}
\END{ALIGN*}
THE MOTIVATION BEHIND THIS CHOICE OF BASIS IS AS FOLLOWS. IT IS WELL-KNOWN THAT $S^6=_2/(3)$ ADMITS A UNIQUE INVARIANT NEARLY K\"{A}HLER STRUCTURE UP TO SIGN [20], AND A DIRECT CALCULATION (BY HAND OR USING COMPUTER ALGEBRA SOFTWARE) SHOWS THAT THE $1$-DIMENSIONAL SPACE OF INVARIANT $2$-FORMS IS SPANNED BY
\[
J= E_1\WEDGE E_2 + E_3\WEDGE E_4 + E_5\WEDGE E_6.
\]
IN PARTICULAR, THE INVARIANT ALMOST COMPLEX STRUCTURES ARE $±J$ AND THE GIVEN BASIS ${E_I}$ IS \EMPH{ADAPTED} TO $J$ IN THE SENSE THAT $J(E_1)=E_2$, $J(E_3)=E_4$, $J(E_5)=E_6$. THIS PROPERTY ALLOWS US TO GIVE AN EASY PROOF OF THE FOLLOWING THEOREM:
\BEGIN{THEOREM} \LABEL{G2SU3THEOREM}
USING THE ABOVE ORTHONORMAL BASIS AND THE CORRESPONDING DESCRIPTION OF THE SPINOR MODULE FROM SECTION \REF{UNDERSTANDINGSPINREP}, THE SPACE OF INVARIANT SPINORS ON $(S^6 = \G_2/\SU(3), G)$ IS GIVEN BY
\BEGIN{ALIGN*}
\SIGMA_{\INV}&= \SPAN_{\C} \{ 1, Y_1\WEDGE Y_2\WEDGE Y_3 \},
\END{ALIGN*}
AND THE SPINORS $\PSI_{\PM}:= 1\PM Y_1\WEDGE Y_2\WEDGE Y_3$ ARE RIEMANNIAN KILLING SPINORS,
\[
\NABLA_X^G \PSI_{\PM} = \PM \FRAC{1}{2\SQRT{3}} X\CDOT \PSI_{\PM}.
\]
\END{THEOREM}
\BEGIN{PROOF}
SINCE THE ORTHONORMAL BASIS $\{E_I\}$ IS ADAPTED TO THE INVARIANT ALMOST COMPLEX STRUCTURE $J$, IT FOLLOWS FROM REMARK \REF{CHANGE_OF_BASIS_EFFECT} THAT $\SIGMA \SIMEQ \LAMBDA^{0,\BULLET}\MATHFRAK{M}$ AS COMPLEX REPRESENTATIONS OF $\SU(3)$, AND THE SPACE OF INVARIANTS IS THEREFORE SPANNED BY $1$ AND THE COMPLEX VOLUME FORM $Y_1\WEDGE Y_2\WEDGE Y_3$ (SEE E.G. \CITE[PROP.\@ F.10]{FULTONHARRIS1991}).
\END{PROOF}
\BEGIN{REMARK}
THE PRECEDING THEOREM CAN ALSO BE PROVED DIRECTLY, BY PERFORMING A SIMILAR CALCULATION AS IN EXAMPLE \REF{SP2SP1}, USING THE OPERATORS
\BEGIN{ALIGN*}
\AD(\NU_1)\RVERT_{\MATHFRAK{M}} &= \FRAC{1}{2}(E_{1,2} - E_{3,4}), \QUAD \AD(\NU_2)\RVERT_{\MATHFRAK{M}}= \FRAC{1}{2}(E_{3,5}+E_{4,6}), \QUAD 	\AD(\NU_3)\RVERT_{\MATHFRAK{M}}= \FRAC{1}{2}( -E_{3,6} + E_{4,5}  ), \\ \AD(\NU_4)\RVERT_{\MATHFRAK{M}} &= \FRAC{1}{2}(E_{1,6} - E_{2,5}  ), \QUAD
\AD(\NU_5)\RVERT_{\MATHFRAK{M}} = \FRAC{1}{2}( -E_{1,5}-E_{2,6}  ), \QUAD \AD(\NU_6)\RVERT_{\MATHFRAK{M}} = \FRAC{1}{2}(-E_{1,4} +E_{2,3}  ), \\ \AD(\NU_7)\RVERT_{\MATHFRAK{M}} &= \FRAC{1}{2} (E_{1,3}+E_{2,4}  ), \QUAD \AD(\NU_8)\RVERT_{\MATHFRAK{M}}= \FRAC{1}{2\SQRT{3}}(-E_{1,2}-E_{3,4} +2E_{5,6}   ).
\END{ALIGN*}
%USING THE SPIN REPRESENTATION DESCRIBED IN SECTION \REF{UNDERSTANDINGSPINREP}, ONE CALCULATES
%	\WIDETILDE{\AD(\NU_1)\RVERT_{\MATHFRAK{M}}}\CDOT \ETA &= \FRAC{1}{4}(E_1\CDOT E_2 + E_3\CDOT E_4)\CDOT \ETA    \\
%	&=\FRAC{1}{4} \LEFT( I(X_1\LRCORNER+Y_1\WEDGE)(Y_1\WEDGE - X_1\LRCORNER) \ETA + I(X_2\LRCORNER+Y_2\WEDGE)(Y_2\WEDGE - X_2\LRCORNER) \ETA\RIGHT)\\
%	&= \FRAC{I}{2}\LEFT( \ETA -Y_1\WEDGE (X_1\LRCORNER\ETA) -Y_2\WEDGE (X_2\LRCORNER \ETA)     \RIGHT)
%FOR ANY $\ETA \IN \SIGMA = \LAMBDA^{\BULLET} L'$. WE THEN HAVE
%	\WIDETILDE{\AD(\NU_1)\RVERT_{\MATHFRAK{M}}}\CDOT 1 &= \FRAC{I}{2}, \QUAD \WIDETILDE{\AD(\NU_1)\RVERT_{\MATHFRAK{M}}}\CDOT Y_1 =0, \QUAD \WIDETILDE{\AD(\NU_1)\RVERT_{\MATHFRAK{M}}}\CDOT Y_2 = 0, \QUAD \WIDETILDE{\AD(\NU_1)\RVERT_{\MATHFRAK{M}}}\CDOT Y_3 = \FRAC{I}{2} Y_3, \\
%	\WIDETILDE{\AD(\NU_1)\RVERT_{\MATHFRAK{M}}}\CDOT (Y_1\WEDGE Y_2) &=  \FRAC{-I}{2} Y_1\WEDGE Y_2 ,  \QUAD \WIDETILDE{\AD(\NU_1)\RVERT_{\MATHFRAK{M}}}\CDOT (Y_2\WEDGE Y_3) =0 , \QUAD
%	\WIDETILDE{\AD(\NU_1)\RVERT_{\MATHFRAK{M}}}\CDOT (Y_1\WEDGE Y_3) = 0 , \\
%	\WIDETILDE{\AD(\NU_1)\RVERT_{\MATHFRAK{M}}}\CDOT (Y_1\WEDGE Y_2\WEDGE Y_3) &=  \FRAC{-I}{2} Y_1\WEDGE Y_2\WEDGE Y_3  ,
%AND HENCE
%\KER \WIDETILDE{\AD(\NU_1)\RVERT_{\MATHFRAK{M}}} = \SPAN_{\C} \{ Y_1,Y_2, Y_2\WEDGE Y_3, Y_1\WEDGE Y_3 \}.
%CONTINUING SIMILARLY FOR THE OTHER ISOTROPY OPERATORS AND TAKING THE INTERSECTION OF THE KERNELS GIVES
%\BIGCAP_{I=1}^8 \KER \WIDETILDE{\AD(\NU_I)\RVERT_{\MATHFRAK{M}}}=\SPAN_{\C} \{ \PSI_{\PM}\}.
\BEGIN{ALIGN*}
\UPLAMBDA^G(E_1)&= \FRAC{1}{2\SQRT{3}}(E_{3,6}+E_{4,5} ),\QUAD
\UPLAMBDA^G(E_2)= \FRAC{1}{2\SQRT{3}}(E_{3,5}-E_{4,6}), \QUAD
\UPLAMBDA^G(E_3)= \FRAC{1}{2\SQRT{3}}(-E_{1,6}-E_{2,5} ) ,\\
\UPLAMBDA^G(E_4)&= \FRAC{1}{2\SQRT{3}}(-E_{1,5}+E_{2,6}), \QUAD
\UPLAMBDA^G(E_5)=\FRAC{1}{2\SQRT{3}}(E_{1,4}+E_{2,3}) ,\QUAD
\UPLAMBDA^G(E_6)=\FRAC{1}{2\SQRT{3}} (E_{1,3}-E_{2,4}).
\END{ALIGN*}
\END{REMARK}
\BEGIN{REMARK}
THE INVARIANT SPINORS $\PSI_{\PM}$ INDUCE A PAIR OF ALMOST COMPLEX STRUCTURES $J_{\PSI_{\PM}}$ VIA
\[
J_{\PSI_{\PM}}(X)\CDOT (\PSI_{\PM})_0 =  IX\CDOT (\PSI_{\PM})_0\QUAD  \TEXT{FOR ALL } X\IN TM,
\]
WHERE $(\PSI_{\PM})_0$ DENOTES THE PROJECTION ONTO THE EVEN HALF-SPINOR MODULE $\SIGMA_0\SUBSETEQ \SIGMA$ (SEE E.G. \CITE[CHAPTER 5.2]{BFGK}); THESE ARE WELL-KNOWN TO BE STRICTLY NEARLY K\"{A}HLER [29], I.E. THEY SATISFY $(\NABLA_XJ_{\PSI_{\PM}})X=0$ FOR ALL $X\IN TM$. INDEED, $(\PSI_+)_0=(\PSI_-)_0=1$, AND ONE FINDS BY A DIRECT CALCULATION IN THE SPIN REPRESENTATION THAT $J_{\PSI_{\PM}}$ COINCIDE WITH THE NEARLY K\"{A}HLER FORM $J$ DEFINED ABOVE. FOR A RECENT HISTORICAL ACCOUNT ON THE NEARLY K\"{A}HLER STRUCTURE ON $S^6$, WE REFER TO [1]; SEE ALSO [2].
\END{REMARK}
NEXT, WE CALCULATE THE AMBROSE-SINGER TORSION AND DETERMINE ITS TYPE:
\BEGIN{PROPOSITION}
THE SPHERE $(S^6=\G_2/\SU(3), G)$ HAS AMBROSE-SINGER TORSION OF TYPE $\MATHCAL{T}_{\TOTALLYSKEW}$ GIVEN BY
\[
T^{\AS} = \FRAC{1}{\SQRT{3}} (-E_{1,3,6} - E_{1,4,5} - E_{2,3,5} + E_{2,4,6}).
\]
\END{PROPOSITION}
\SUBSECTION{$S^7=\SPIN(7)/\G_2$}
THE ISOTROPY REPRESENTATION IS AGAIN IRREDUCIBLE, SO WE USE THE INVARIANT METRIC INDUCED BY THE INNER PRODUCT $G=B_0$ ON $𝔰𝔭𝔦𝔫(7)$, AND CHOOSE THE REDUCTIVE COMPLEMENT $𝔪:=(𝔤_2)^⊥$ WITH RESPECT TO THIS INNER PRODUCT. FOLLOWING THE NOTATION OF PROPOSITION \REF{ONBASISSPIN7SPIN9}, WE HAVE
\BEGIN{ALIGN*}
\MATHFRAK{SPIN}(7) &= \{\NU_1,\DOTS, \NU_{14}, \NU'_{15},\DOTS ,\NU'_{21}\}, \QUAD
\MATHFRAK{G}_2 = \{ \NU_1,\DOTS \NU_{14}\},
\END{ALIGN*}
\BEGIN{ALIGN*}
\MATHFRAK{M}&= \{E_1,\DOTS, E_7\}, \ \ \TEXT{WHERE } E_I:= \NU_{14+I}.
\END{ALIGN*}
THE FOLLOWING THEOREM DESCRIBING THE INVARIANT SPINORS IN THIS CASE MAY BE COMPARED TO \CITE[\S3 CASE 4]{WANG}, WHERE HE GIVES AN EXPRESSION FOR THE PARALLEL SPINOR ON A $_2$-MANIFOLD.
\BEGIN{THEOREM}\LABEL{G2SPIN7_INV_SPINORS_THEOREM} USING THE ABOVE ORTHONORMAL BASIS AND THE CORRESPONDING DESCRIPTION OF THE SPINOR MODULE FROM REMARK \REF{REORDEREDCLIFFALGREPRESENTATION}, THE SPACE OF INVARIANT SPINORS ON $(S^7= \SPIN(7)/\G_2, G)$ IS GIVEN BY
$$\SIGMA_{\INV} = \SPAN_{\C}\{  -1+Y_1\WEDGE Y_2\WEDGE Y_3 \}  , $$
AND THE SPINOR $\PSI := -1+Y_1\WEDGE Y_2\WEDGE Y_3$ IS A RIEMANNIAN KILLING SPINOR,
\[
\NABLA_X^G \PSI = \FRAC{\SQRT{3}}{4\SQRT{2}} X \CDOT \PSI.
\]
\END{THEOREM}
\BEGIN{PROOF}
THIS FOLLOWS FROM THE SAME CALCULATION AS IN EXAMPLE \REF{SP2SP1}, USING THE OPERATORS {\SMALL
\BEGIN{ALIGN*}
\AD(\NU_1)\RVERT_{\MATHFRAK{M}}&= \FRAC{1}{2}(E_{2,3}-E_{6,7}), \QUAD \AD(\NU_2)\RVERT_{\MATHFRAK{M}}= \FRAC{1}{2}(-E_{2,4}-E_{3,5} ), \QUAD \AD(\NU_3)\RVERT_{\MATHFRAK{M}}= \FRAC{1}{2}(-E_{2,5}+E_{3,4}),\\
\AD(\NU_4)\RVERT_{\MATHFRAK{M}}&= \FRAC{1}{2}(E_{4,7}-E_{5,6}), \QUAD \AD(\NU_5)\RVERT_{\MATHFRAK{M}} = \FRAC{1}{2}(-E_{4,6}-E_{5,7}), \QUAD \AD(\NU_6)\RVERT_{\MATHFRAK{M}}= \FRAC{1}{2}( E_{2,7}-E_{3,6}),\\
\AD(\NU_7)\RVERT_{\MATHFRAK{M}}&= \FRAC{1}{2}(-E_{2,6}-E_{3,7}),\QUAD \AD(\NU_8)\RVERT_{\MATHFRAK{M}} = \FRAC{1}{2\SQRT{3}}(E_{2,3} -2E_{4,5} +E_{6,7}), \QUAD
\AD(\NU_9)\RVERT_{\MATHFRAK{M}}= \FRAC{1}{2\SQRT{3}}( 2E_{1,7} -E_{2,5} -E_{3,4}), \\ \AD(\NU_{10})\RVERT_{\MATHFRAK{M}} &= \FRAC{1}{2\SQRT{3}}( 2E_{1,6} +E_{2,4} -E_{3,5}), \QUAD
\AD(\NU_{11})\RVERT_{\MATHFRAK{M}}= \FRAC{1}{2\SQRT{3}}( -2E_{1,3} +E_{4,7} +E_{5,6} ),\\ \AD(\NU_{12})\RVERT_{\MATHFRAK{M}} &= \FRAC{1}{2\SQRT{3}}( 2E_{1,2} +E_{4,6} -E_{5,7}), \QUAD
\AD(\NU_{13})\RVERT_{\MATHFRAK{M}} = \FRAC{1}{2\SQRT{3}}(  2E_{1,5} +E_{2,7}+E_{3,6} ), \\ \AD(\NU_{14})\RVERT_{\MATHFRAK{M}}&= \FRAC{1}{2\SQRT{3}}(-2E_{1,4} +E_{2,6} -E_{3,7}   ),
\END{ALIGN*}
\BEGIN{ALIGN*}
\UPLAMBDA^G(E_1) &= \FRAC{1}{2\SQRT{6}}(E_{2,3}+E_{4,5}+E_{6,7}), \QUAD \UPLAMBDA^G(E_2) = \FRAC{1}{2\SQRT{6}}(-E_{1,3}-E_{4,7} -E_{5,6}),\QUAD  \UPLAMBDA^G(E_3) = \FRAC{1}{2\SQRT{6}}(E_{1,2}-E_{4,6} +E_{5,7}),\\
\UPLAMBDA^G(E_4) &= \FRAC{1}{2\SQRT{6}}(-E_{1,5}+E_{2,7} +E_{3,6} ),\QUAD
\UPLAMBDA^G(E_5) = \FRAC{1}{2\SQRT{6}}(E_{1,4}+E_{2,6} -E_{3,7}),\QUAD
\UPLAMBDA^G(E_6) =\FRAC{1}{2\SQRT{6}}(-E_{1,7}-E_{2,5} -E_{3,4} ),\\
\UPLAMBDA^G(E_7) &= \FRAC{1}{2\SQRT{6}}(E_{1,6} -E_{2,4} +E_{3,5} ).
\END{ALIGN*}
\END{PROOF}
%	IT IS WELL-KNOWN THAT A UNIT LENGTH SPINOR $\VARPHI$ ON A 7-DIMENSIONAL MANIFOLD INDUCES A $\G_2$-STRUCTURE ON THE MANIFOLD VIA THE 3-FORM $\OMEGA_{\VARPHI}(X,Y,Z) := \LANGLE X\CDOT Y\CDOT Z\CDOT \VARPHI, \VARPHI\RANGLE$, WHERE $\LANGLE\CDOT,\CDOT\RANGLE$ DENOTES THE STANDARD HERMITIAN INNER PRODUCT ON THE SPINOR BUNDLE (SEE E.G. [2]). BY RESCALING THE CONSTANT LENGTH INVARIANT SPINOR $\PSI$ DEFINED ABOVE TO HAVE UNIT LENGTH, ONE THUS OBTAINS A $\G_2$-STRUCTURE, AND SINCE $\PSI$ IS A RIEMANNIAN KILLING SPINOR THE RESULTING $\G_2$-STRUCTURE IS NEARLY PARALLEL [25].
NEXT, WE CALCULATE THE AMBROSE-SINGER TORSION AND DETERMINE ITS TYPE:
\BEGIN{PROPOSITION}
THE SPHERE $(S^7=\SPIN(7)/\G_2, G)$ HAS AMBROSE-SINGER TORSION OF TYPE $\MATHCAL{T}_{\TOTALLYSKEW}$ GIVEN BY
\[
T^{\AS} = \FRAC{1}{\SQRT{6}} ( - E_{1,2,3} - E_{1,4,5} - E_{1,6,7} + E_{2,4,7} + E_{2,5,6} + E_{3,4,6} - E_{3,5,7}).
\]
\END{PROPOSITION}
\SUBSECTION{$S^{15}=\SPIN(9)/\SPIN(7)$}\LABEL{SPIN9SPIN7}
THE ISOTROPY REPRESENTATION IN THIS CASE SPLITS INTO TWO NON-EQUIVALENT MODULES (ONE COPY OF THE SPIN REPRESENTATION AND ONE COPY OF THE STANDARD REPRESENTATION), HENCE THERE IS A TWO-DIMENSIONAL FAMILY OF INVARIANT METRICS. INDEED, INCLUDE $𝔰𝔭𝔦𝔫(7)⊆𝔰𝔭𝔦𝔫(9)$ AS IN PROPOSITION \REF{ONBASISSPIN7SPIN9} AND SET $𝔪:=(𝔰𝔭𝔦𝔫(7))^⊥$, WHERE ORTHOGONALITY IS TAKEN WITH RESPECT TO THE KILLING FORM ON $𝔰𝔭𝔦𝔫(9)$. EXPLICITLY, IN THE NOTATION OF PROPOSITION \REF{ONBASISSPIN7SPIN9},
\BEGIN{ALIGN*}
\MATHFRAK{SPIN}(9)&= \SPAN_{\R} \{\IOTA(\NU_1),\DOTS,\IOTA(\NU_{14}), \IOTA(\NU'_{15}),\DOTS, \IOTA(\NU'_{21}), \NU'_{22},\DOTS, \NU'_{36}     \}, \\
\MATHFRAK{SPIN}(7)&= \SPAN_{\R} \{ \NU'_{22},\DOTS, \NU'_{36} \},
\END{ALIGN*}
\BEGIN{ALIGN*}
\MATHFRAK{M}&= \SPAN_{\R} \{ \WIDEHAT{E}_1,\DOTS, \WIDEHAT{E}_{15}\}, \ \ \TEXT{WHERE } \WIDEHAT{E}_I:= \NU'_{21+I}.
\END{ALIGN*}
THE TWO IRREDUCIBLE ISOTROPY SUMMANDS ARE GIVEN BY
\BEGIN{ALIGN*}
\MATHFRAK{M}_F&:= \SPAN_{\R} \{ \WIDEHAT{E}_1,\DOTS ,\WIDEHAT{E}_7\},\QUAD
\MATHFRAK{M}_B:= \SPAN_{\R} \{ \WIDEHAT{E}_8,\DOTS, \WIDEHAT{E}_{15} \},
\END{ALIGN*}
CORRESPONDING TO THE TANGENT SPACES OF THE FIBER AND BASE RESPECTIVELY OF THE OCTONIONIC HOPF FIBRATION,
\BEGIN{ALIGN}\LABEL{OCTHOPF}
S^{7}=\FRAC{\SPIN(8)}{\SPIN(7)} \HOOKRIGHTARROW S^{15}=\FRAC{\SPIN(9)}{\SPIN(7)} \TO S^8=\FRAC{\SPIN(9)}{\SPIN(8)} .
\END{ALIGN}
THE TWO-DIMENSIONAL FAMILY OF INVARIANT METRICS IS PARAMETERIZED BY
\[
G_{A,B}:= AB_0\RVERT_{\MATHFRAK{M}_F\TIMES \MATHFRAK{M}_F} + B B_0\RVERT_{\MATHFRAK{M}_B\TIMES \MATHFRAK{M}_B} , \QUAD A,B>0
\]
AND A $G_A,B$-ORTHONORMAL BASIS OF $𝔪$ IS GIVEN BY ${E_I}_I=1^15$, WHERE
\BEGIN{ALIGN}\LABEL{S15ONB}
E_I:= \BEGIN{CASES*}
\FRAC{1}{\SQRT{A}} \WIDEHAT{E}_I & IF $I=1,\DOTS, 7$, \\
\FRAC{1}{\SQRT{B}}\WIDEHAT{E}_I   & IF $I=8,\DOTS, 15$.
\END{CASES*}
\END{ALIGN}
BY MANUALLY CHECKING THE SECTIONAL CURVATURES, ONE FINDS THAT THE ROUND METRIC CORRESPONDS TO THE PARAMETERS $A=1/2$, $B=1/8$ AND WE RECALL (SEE E.G. [46]) THAT THE ROUND METRIC IS THE ONLY MEMBER OF THE FAMILY $G_A,B$ WHICH CAN BE ISOMETRIC TO ONE OF THE METRICS FOR $G=(N+1)$, $(N+1)$, $(N)$, OR $(N)(1)$ CONSIDERED IN PREVIOUS SECTIONS. IN THE FOLLOWING PROPOSITION WE GIVE A COMPLETE DESCRIPTION OF THE INVARIANT DIFFERENTIAL FORMS:
\BEGIN{PROPOSITION}\LABEL{S15INVARIANTFORMS}
UP TO TAKING HODGE DUALS, THE INVARIANT DIFFERENTIAL FORMS ON $(S^{15}=\SPIN(9)/\SPIN(7), G_{A,B})$ ARE GIVEN IN TABLE \REF{TAB:INVARIANT_FORMS_S15}, WHERE $\OMEGA \IN \LAMBDA^3\MATHFRAK{M}$, $\PSI\IN\LAMBDA^4\MATHFRAK{M}$ AND THEIR EXTERIOR DERIVATIVES ARE DEFINED BY THE FORMULAS IN APPENDIX \REF{APPENDIXB_FORMS}, AND $\TEXT{\EMPH{PR}}_{I,J}$ DENOTES THE PROJECTION ONTO $\LAMBDA^I\MATHFRAK{M}_F\OTIMES \LAMBDA^J\MATHFRAK{M}_B$.
\END{PROPOSITION}
\BEGIN{TABLE}[H!]
\CENTERING
\CAPTION{INVARIANT DIFFERENTIAL FORMS ON $(S^{15}=\SPIN(9)/\SPIN(7), G_{A,B})$}
\BEGIN{TABULAR}{|L||L|L|}\HLINE
$K$  & $\DIM \LAMBDA^K_{\INV}$ & BASIS FOR $ \LAMBDA^K_{\INV}$ \\ \HLINE
$0$    & $1$ & $1$  \\
$1$ & $0$ & $0$  \\
$2$ & $0$ & $0$ \\
$3$   & $1$ & $\OMEGA $   \\
$4$  & $2$   & $D\OMEGA, \PSI$  \\
$5$ &  $1$  &   $D\PSI$  \\
$6$ &  $0$   &  $0$  \\
$7$ &   $4$  &   $\TEXT{PR}_{1,6}(\OMEGA\WEDGE D\OMEGA),\ \TEXT{PR}_{3,4}(\OMEGA\WEDGE D\OMEGA),\ \AST(\OMEGA \WEDGE D\PSI),\ \AST(\PSI\WEDGE \PSI )$   \\ \HLINE
\END{TABULAR}
\LABEL{TAB:INVARIANT_FORMS_S15}
\END{TABLE}
IN PARTICULAR, THIS SHOWS THAT THE INVARIANT DIFFERENTIAL FORMS ON $S^15=(9)/(7)$ ARE GENERATED BY $Ω$ AND $Ψ$ AND THEIR DERIVATIVES, AND IN WHAT FOLLOWS WE SHALL EXAMINE MORE CLOSELY THE GEOMETRIC AND SPINORIAL FEATURES OF THESE FORMS.
\BEGIN{REMARK} THE 4-FORM $\PSI$ IS PURELY HORIZONTAL, I.E. $\PSI \IN \LAMBDA^4\MB$, AND THE HORIZONTAL COMPONENT OF $D\OMEGA$ IS A MULTIPLE OF $\PSI$. ONE FINDS THAT $\PSI$ IS \EMPH{NOT} INVARIANT FOR THE LARGER GROUP $\SPIN(8)$, HENCE IT DOES NOT DESCEND TO AN INVARIANT 4-FORM ON THE BASE SPACE $S^8 = \SPIN(9)/\SPIN(8)$ OF THE OCTONIONIC HOPF FIBRATION (\REF{OCTHOPF}).
\END{REMARK}
\BEGIN{REMARK}\LABEL{UWE2}
NOTE THAT THE EXISTENCE OF AN INVARIANT SPINOR AND AN INVARIANT $4$-FORM ON
$S^{15}$ IS CLEAR BY THE FOLLOWING ARGUMENT. IN THIS CASE THE ISOTROPY REPRESENTATION IS REALIZED AS A DIRECT SUM OF THE STANDARD
AND SPIN REPRESENTATIONS $\MATHBB{R}^{15} \CONG \MATHBB{R}^7 \OPLUS \DELTA_7$.
THIS DECOMPOSITION OF $\SPIN(7)$-REPRESENTATIONS IS ORTHOGONAL WITH RESPECT TO THE FORM. THUS IT INDUCES A HOMOMORPHISM $\VARPHI: \SPIN(7) \TO \SO(7) \TIMES \SO(8)$, WHICH LIFTS TO $\SPIN(7) \TO \SPIN(7) \TIMES \SPIN(8) \SUBSETEQ \SPIN(15)$.

IT IS KNOWN THAT THERE IS AN INVARIANT VECTOR IN $\LAMBDA^4 \DELTA_7$ (CORRESPONDING TO THE $4$-FORM $\PSI$ IN TABLE \REF{TAB:INVARIANT_FORMS_S15}). BY THE ABOVE, $\LAMBDA^4 \DELTA_7 \SUBSETEQ \LAMBDA^4 \DELTA_{15}$. THEREFORE THERE IS ALSO AN INVARIANT VECTOR IN $\LAMBDA^4 \DELTA_{15} |_{\VARPHI}$, I.E., A FOUR-FORM ON $\MATHBB{S}^{15}$ INVARIANT UNDER $\SPIN(7)$.

IN THIS SITUATION, $\DELTA_{15}|_{\SPIN(7) \TIMES \SPIN(8)} \CONG \DELTA_7 \OTIMES \DELTA_8$. WE NOW RESTRICT THIS TO A REPRESENTATION OF $\SPIN(7)$ VIA $\VARPHI$.  NOTE THAT $\VARPHI(G)=(G,\RHO(G))$ WHERE $\RHO: \SPIN(7) \TO \SPIN(8)$ IS THE LIFT OF THE SPIN REPRESENTATION TO $\SPIN(8)$.  THANKS TO TRIALITY (SEE [45], (III) ON P.~61), WE CAN DECOMPOSE
\[
\DELTA_8|_{\RHO} \CONG (\DELTA_8^+ \OPLUS \DELTA_8^-)|_{\RHO} \CONG \DELTA_7 \OPLUS \R^7 \OPLUS \R.
\]
FINALLY, TENSORING BY $\DELTA_7$, WE GET:
\[
\DELTA_{15}|_{\VARPHI} \CONG (\DELTA_7 \OTIMES \DELTA_7) \OPLUS (\DELTA_7 \OTIMES \R^7) \OPLUS \DELTA_7.
\]
SINCE $\DELTA_7, \R^7$, AND $\R$ ARE REAL IRREDUCIBLE REPRESENTATIONS, A TENSOR PRODUCT OF TWO OF THESE HAS AN INVARIANT VECTOR IF AND ONLY IF THE TWO ARE THE SAME REPRESENTATION.  SO THERE IS A UNIQUE INVARIANT SPINOR, CORRESPONDING TO THE INVARIANT ELEMENT OF $\DELTA_7 \OTIMES \DELTA_7$ WHICH EXPRESSES THE INVARIANT INNER PRODUCT ON $\DELTA_7$.
\END{REMARK}
IN THE FOLLOWING THEOREM WE DESCRIBE THE SPACE OF INVARIANT SPINORS:
\BEGIN{THEOREM}
USING THE ABOVE $G_{A,B}$-ORTHONORMAL BASIS AND THE CORRESPONDING DESCRIPTION OF THE SPINOR MODULE FROM REMARK \REF{REORDEREDCLIFFALGREPRESENTATION}, THE SPACE OF INVARIANT SPINORS ON $(S^{15}=\SPIN(9)/\SPIN(7),G_{A,B})$ IS GIVEN BY
\[
\SIGMA_{\INV} = \SPAN_{\C} \{ \PSI \},
\]
%	\BEGIN{ALIGN*}
%	\PSI&:= \FRAC{1}{2} Y_{1,5}+\FRAC{1}{2} Y_{1,7}+\FRAC{1}{2} Y_{3,5}+\FRAC{1}{2} Y_{3,7}+ Y_{1,2,3}-\FRAC{1}{2} Y_{1,2,5}+\FRAC{1}{2} Y_{1,2,7}-Y_{1,3,6}-\FRAC{1}{2} Y_{1,4,5} \\
%	&\QQUAD   -\FRAC{1}{2} Y_{1,4,7}+\FRAC{1}{2} Y_{1,5,6}-\FRAC{1}{2} Y_{1,6,7}+\FRAC{1}{2} Y_{2,3,5}-\FRAC{1}{2} Y_{2,3,7}+Y_{2,5,7}-\FRAC{1}{2} Y_{3,4,5}-\FRAC{1}{2} Y_{3,4,7} \\ &\QQUAD    -\FRAC{1}{2} Y_{3,5,6}+\FRAC{1}{2} Y_{3,6,7}-Y_{5,6,7}+Y_{1,2,3,4}-\FRAC{1}{2} Y_{1,2,4,5}+\FRAC{1}{2} Y_{1,2,4,7}+\FRAC{1}{2} Y_{1,2,5,6} \\ &\QQUAD   +\FRAC{1}{2} Y_{1,2,6,7}-Y_{1,3,4,6}+\FRAC{1}{2} Y_{1,4,5,6}-\FRAC{1}{2} Y_{1,4,6,7}+\FRAC{1}{2} Y_{2,3,4,5}-\FRAC{1}{2} Y_{2,3,4,7}+\FRAC{1}{2} Y_{2,3,5,6} \\ &\QQUAD   +\FRAC{1}{2} Y_{2,3,6,7}+Y_{2,4,5,7}-\FRAC{1}{2} Y_{3,4,5,6}+\FRAC{1}{2} Y_{3,4,6,7}-Y_{4,5,6,7}-\FRAC{1}{2} Y_{1,2,4,5,6}-\FRAC{1}{2} Y_{1,2,4,6,7}   \\   &\QQUAD   -\FRAC{1}{2} Y_{2,3,4,5,6}-\FRAC{1}{2} Y_{2,3,4,6,7},
%	\END{ALIGN*}
\[
\PSI := \FRAC{1}{2\SQRT{2}}( -IY_{1,5} + Y_{1,2,3} +Y_{2,5,7} -Y_{3,5,6} +IY_{1,2,4,7} -IY_{1,3,4,6} -IY_{4,5,6,7} + Y_{2,3,4,6,7}  ),
\]
AND WE USE THE NOTATION $Y_{I_1,\DOTS,I_P}:= Y_{I_1}\WEDGE \DOTS \WEDGE Y_{I_P}$. IT SATISFIES THE SPINORIAL EQUATION
\BEGIN{ALIGN} \LABEL{SPIN9SPIN7SPINORIALEQN}
\NABLA^G_{E_I} \PSI = \BEGIN{CASES*}
\LEFT(\FRAC{A-2 B}{40 B \SQRT{2A} } \RIGHT)   \OMEGA\CDOT E_I\CDOT \PSI & IF $I=1,\DOTS, 7$, \\ \LEFT(  \FRAC{A+2 B}{16 B \SQRT{2}  (A-4 B)}\RIGHT) \OMEGA \CDOT E_I\CDOT \PSI +   \LEFT( \FRAC{\SQRT{A}}{16 (A-4 B)} \RIGHT) D\OMEGA \CDOT E_I\CDOT \PSI    & IF $I=8,\DOTS, 15$.
\END{CASES*}
\END{ALIGN}
\END{THEOREM}
\BEGIN{PROOF}
FOLLOWS THE SAME PROCEDURE AS THE PROOF OF THEOREM \REF{G2SPIN7_INV_SPINORS_THEOREM}. TO DIFFERENTIATE $\PSI$, WE NOTE THAT THE NOMIZU MAP OF THE LEVI-CIVITA CONNECTION IS GIVEN BY
\[
\UPLAMBDA^G(X)Y  = \BEGIN{CASES*}
\FRAC{1}{2}[X,Y]_{\MATHFRAK{M}} & IF $X,Y\IN\MF $, \\
(1-\FRAC{A}{2B}) [X,Y]_{\MATHFRAK{M}} & IF $X\IN\MF$, $Y\IN\MB$, \\
\FRAC{1}{2B}  [X,Y]_{\MATHFRAK{M}} & IF $X\IN\MB$, $Y\IN\MF$,\\
\FRAC{1}{2} [X,Y]_{\MATHFRAK{M}} & IF $X,Y\IN \MB$.
\END{CASES*}
\]
\END{PROOF}
THE FACT THAT THE SPINORIAL EQUATION (\REF{SPIN9SPIN7SPINORIALEQN}) DEPENDS ONLY ON THE INVARIANT 3-FORM $Ω$ SUGGESTS THAT THERE IS AN INTRINSIC RELATIONSHIP BETWEEN $Ψ$ AND $Ω$. INDEED, THE SPINOR $Ψ$ DETERMINES $Ω$ VIA THE \EMPH{SQUARING CONSTRUCTION}:
\BEGIN{ALIGN*}
\OMEGA(X,Y,Z) &= -2\LANGLE (X\WEDGE Y \WEDGE Z) \CDOT \PSI, \PSI \RANGLE \QUAD \TEXT{FOR ALL } X,Y,Z\IN TM,
\END{ALIGN*}
WHERE $⟨·, ·⟩$ DENOTES THE USUAL HERMITIAN INNER PRODUCT ON THE SPINOR BUNDLE. CONVERSELY, (\REF{SPIN9SPIN7SPINORIALEQN}) SHOWS THAT $Ω$ DETERMINES $Ψ$ UP TO FIRST ORDER. IN GENERAL, FOR EACH INTEGER $K≥0$ THE SQUARING CONSTRUCTION DETERMINES AN INVARIANT $K$-FORM $Ω_(K)$ VIA
\[
\OMEGA_{(K)}(X_1,\DOTS,X_K) := \RE \LANGLE (X_1\WEDGE \DOTS\WEDGE X_K)\CDOT \PSI,\PSI\RANGLE \QUAD \TEXT{FOR ALL } X_1,\DOTS X_K\IN TM,
\]
AND ONE CAN ASK WHETHER $Ψ$ IS RELATED TO OTHER INVARIANT FORMS FROM TABLE \REF{TAB:INVARIANT_FORMS_S15} BY THIS CONSTRUCTION. WE FIND:
\BEGIN{PROPOSITION}
UP TO TAKING HODGE DUALS, THE DIFFERENTIAL FORMS OBTAINED FROM THE INVARIANT SPINOR $\PSI$ VIA THE SQUARING CONSTRUCTION ARE GIVEN IN TABLE \REF{TAB:S15SQUARINGCONSTRUCTION}.
\END{PROPOSITION}
\BEGIN{TABLE}[H!]
\CENTERING
\CAPTION{FORMS ON $(S^{15}=\SPIN(9)/\SPIN(7),G_{A,B})$ OBTAINED FROM $\PSI$ VIA THE SQUARING CONSTRUCTION}
\BEGIN{TABULAR}{|L||L|}\HLINE
$K$  &  $ \OMEGA_{(N)}$ \\ \HLINE
$0$    & $1$  \\
$1$ & $0$   \\
$2$ & $0$  \\
$3$   & $\OMEGA$    \\
$4$  & $-\FRAC{\SQRT{A}}{2\SQRT{2}} \TEXT{PR}_{2,2}(D\OMEGA)$    \\
$5$ &  $0$   \\
$6$ &  $0$   \\
$7$ &   $-\FRAC{B\SQRT{2}}{18\SQRT{A}} \TEXT{PR}_{1,6}(\OMEGA\WEDGE D\OMEGA) + \FRAC{\SQRT{A}}{6\SQRT{2}} \TEXT{PR}_{3,4}(\OMEGA\WEDGE D\OMEGA) -\FRAC{\SQRT{A}}{8\SQRT{2}} \AST(\OMEGA\WEDGE D\PSI) - \FRAC{1}{14} \AST(\PSI\WEDGE \PSI)$     \\ \HLINE
\END{TABULAR}
\LABEL{TAB:S15SQUARINGCONSTRUCTION}
\END{TABLE}
\BEGIN{PROOF}NOTE THAT INVARIANCE OF $\PSI$ IMPLIES THAT EACH $\OMEGA_{(N)}$ IS INVARIANT TOO, AND THUS PROPOSITION \REF{S15INVARIANTFORMS} GREATLY LIMITS THE POSSIBILITIES FOR $\OMEGA_{(N)}$. ONE IMMEDIATELY SEES THAT $\OMEGA_{(1)}=\OMEGA_{(2)}=\OMEGA_{(6)} =0$, AND THE CASE $K=3$ IS GIVEN ABOVE. EXAMINING NEXT THE CASE OF $4$-FORMS, WE CONSIDER THE PROJECTION
\[
\TEXT{PR}_{2,2}(D\OMEGA) = D\OMEGA + \FRAC{3\SQRT{A}}{B\SQRT{2}}\PSI
\]
OF THE INVARIANT $4$-FORM $D\OMEGA$ ONTO $\LAMBDA^2\MATHFRAK{M}_F\OTIMES \LAMBDA^2\MATHFRAK{M}_B$, AND IT IS EASY TO SEE THAT $\{\PSI, \ \TEXT{PR}_{2,2}(D\OMEGA)\}$ IS A BASIS FOR THE SPACE OF INVARIANT $4$-FORMS. WRITING $\OMEGA_{(4)}=\LAMBDA_{1}\PSI + \LAMBDA_{2}\TEXT{PR}_{2,2}(D\OMEGA)$, A STRAIGHTFORWARD CALCULATION IN THE SPIN REPRESENTATION SHOWS THAT \[\OMEGA_{(4)}(E_8,E_9,E_{10},E_{11}) = 0 ,\QQUAD \OMEGA_{(4)}(E_1,E_2,E_{8},E_{11})=-\FRAC{1}{2},\]
HENCE $\LAMBDA_{1}=0$ AND $\LAMBDA_{2}=-\FRAC{\SQRT{A}}{2\SQRT{2}}$ (BY COMPARING WITH THE FORMULAS IN APPENDIX \REF{APPENDIXB_FORMS})
SIMILARLY, FOR $5$-FORMS, ONE CALCULATES
\[
\OMEGA_{(5)}(E_1,E_8,E_{10},E_{12},E_{15}) = 0 \NEQ \SQRT{\FRAC{2}{A}} = D\PSI (E_1,E_8,E_{10},E_{12},E_{15}),
\]
AND WE CONCLUDE FROM TABLE \REF{TAB:INVARIANT_FORMS_S15} THAT $\OMEGA_{(5)}=0$. IN DEGREE $7$, WE WRITE
\[
\OMEGA_{(7)}= \LAMBDA_1 \TEXT{PR}_{1,6}(\OMEGA\WEDGE D\OMEGA) + \LAMBDA_2 \TEXT{PR}_{3,4}(\OMEGA\WEDGE D\OMEGA) + \LAMBDA_3(\AST(\OMEGA \WEDGE D\PSI)) + \LAMBDA_4(\AST(\PSI\WEDGE \PSI))
\]
AS A LINEAR COMBINATION OF THE $7$-FORMS FROM TABLE \REF{TAB:INVARIANT_FORMS_S15}. A STRAIGHTFORWARD CALCULATION GIVES
\BEGIN{ALIGN*}
\OMEGA_{(7)}(E_1,E_8,E_9,E_{10},E_{11},E_{12},E_{13}) &=  \FRAC{1}{2}, \QQUAD \OMEGA_{(7)}(E_1,E_2,E_3,E_8,E_9,E_{10},E_{11})= 1\\
\OMEGA_{(7)}(E_3,E_4,E_5,E_6,E_7,E_{13},E_{14}) &= \FRAC{1}{2} , \QQUAD \OMEGA_{(7)}(E_1,E_2,E_3,E_4,E_5,E_6,E_7) = -1 ,
\END{ALIGN*}
AND IT THEN FOLLOWS FROM THE FORMULAS IN APPENDIX \REF{APPENDIXB_FORMS} THAT
\[
\LAMBDA_1 = -\FRAC{B\SQRT{2}}{18\SQRT{A}}, \QUAD \LAMBDA_2 =\FRAC{\SQRT{A}}{6\SQRT{2}} , \QUAD \LAMBDA_3= -\FRAC{\SQRT{A}}{8\SQRT{2}}, \QUAD \LAMBDA_4=-\FRAC{1}{14} .
\]
FOR HIGHER DEGREES, WE NOTE THAT THE SQUARING CONSTRUCTION IS COMPATIBLE WITH THE HODGE STAR IN THE FOLLOWING SENSE: FOR ANY MULTI-INDEX $I=\{I_1,\DOTS,I_K\}$, DEFINE A MULTI-VECTOR $ E_I:=E_{I_1}\WEDGE \DOTS \WEDGE E_{I_K}$, AND RECALL THAT $\AST E_I =\TEXT{SIGN}(I\CUP \WIDEHAT{I}) E_{\WIDEHAT{I}}$, WHERE $\WIDEHAT{I}=\{1,2,\DOTS, 15\}\SETMINUS I$ (WITH UNION AND COMPLEMENT TAKEN AS ORDERED SETS). SINCE $E_I\WEDGE (E_{\WIDEHAT{I}})=\TEXT{SIGN}(I\CUP \WIDEHAT{I}) E_1\WEDGE E_2\WEDGE \DOTS \WEDGE E_{15}$ ACTS ON THE SPIN REPRESENTATION BY $\TEXT{SIGN}(I\CUP \WIDEHAT{I}) \ID$, WE THEN HAVE
\BEGIN{ALIGN*}
\OMEGA_{(K)}(E_I)&= \RE \LANGLE   E_{I}\CDOT \PSI, \PSI\RANGLE = \TEXT{SIGN}(I\CUP \WIDEHAT{I}) \ \RE \LANGLE   E_{I}\CDOT E_I\CDOT E_{\WIDEHAT{I}}\CDOT \PSI, \PSI\RANGLE = (-1)^{\FRAC{K(K+1)}{2}}\TEXT{SIGN}(I\CUP \WIDEHAT{I})  \ \RE \LANGLE  E_{\WIDEHAT{I}}\CDOT \PSI, \PSI\RANGLE \\
&= (-1)^{\FRAC{K(K+1)}{2}} \OMEGA_{(15-K)}(\AST E_I),
\END{ALIGN*}
AND IT FOLLOWS THAT $\OMEGA_{(15-K)} = (-1)^{\FRAC{K(K+1)}{2}} \AST\OMEGA_{(K)}$. IN PARTICULAR, THE FORMS $\OMEGA_{(K)}$ WITH $K\GEQ 8$ ARE DETERMINED BY THOSE IN TABLE \REF{TAB:S15SQUARINGCONSTRUCTION}.
\END{PROOF}
%THE EXISTENCE OF THE INVARIANT HORIZONTAL 4-FORM $\PSI \IN \WEDGE^4 \MB$ IS ALSO EXPECTED, SINCE THE ISOTROPY GROUP $\SPIN(7)$ ACTS ON $\MB$ BY THE SPIN REPRESENTATION, AND THIS REPRESENTATION IS KNOWN TO STABILIZE A 4-FORM (SEE \CITE{HARVEYLAWSON82}, FOR EXAMPLE).
FINALLY, WE CALCULATE THE AMBROSE-SINGER TORSION AND DETERMINE ITS TYPE:
\BEGIN{PROPOSITION}
FOR ANY $A,B>0$ THE SPHERE $(S^{15}= \SPIN(9)/\SPIN(7), G_{A,B})$ HAS AMBROSE-SINGER TORSION OF TYPE $\MATHCAL{T}_{\TOTALLYSKEW}\OPLUS \MATHCAL{T}_{\CT}$, GIVEN BY {\SMALL
\BEGIN{ALIGN*}
T^{\AS}(E_I,-) &=\FRAC{1}{2\SQRT{2A}} E_I\LRCORNER \OMEGA \QUAD  (I=1,\DOTS, 7),\QUAD
T^{\AS}(E_I,-) = \OMEGA_I' \QUAD (I=8,\DOTS,15),
\END{ALIGN*}
WHERE $\OMEGA_I'$ ARE THE $(0,2)$-TENSORS DETERMINED BY
\[
\OMEGA_I'(E_J,E_K) = \BEGIN{CASES*}
\FRAC{1}{2\SQRT{2A}}  \OMEGA(E_I,E_J,E_K)  & IF $J<K$, \\ \FRAC{\SQRT{A}}{2B\SQRT{2}} \OMEGA(E_I,E_J,E_K)  & IF $J>K$, \\
0 & IF $J=K$.
\END{CASES*}
\]
THE PROJECTION OF $T^{\AS}$ ONTO $\MATHCAL{T}_{\TOTALLYSKEW}$ IS
\[
T^{\AS}_{\TOTALLYSKEW} = \FRAC{(A+2B)}{6B\SQRT{2A}} \OMEGA,
\]
WITH $T^{\AS}= T^{\AS}_{\TOTALLYSKEW}$ IF AND ONLY IF $A=B$.
\END{PROPOSITION}

\SECTION{GENERALIZED KILLING SPINORS ON ROUND SPHERES}
IN THIS SECTION WE REVISIT OUR FINDINGS FOR THE ROUND METRIC IN EACH CASE, AND COMPARE WITH THE KNOWN RESULTS OF MOROIANU AND SEMMELMANN [39, 40]. IN SECTION 4 OF [40] IT IS SHOWN THAT, FOR THE ROUND SPHERE $S^N$ ($N≥3$), IF THERE EXISTS A GENERALIZED KILLING SPINOR WITH EXACTLY TWO EIGENVALUES, THEN THOSE EIGENVALUES ARE EQUAL TO $1/2$, $-3/2$ (UP TO A CHANGE OF ORIENTATION) AND THEY OCCUR WITH MULTIPLICITY $M_1/2$ AND $M_-3/2= M_1/2 +1$ RESPECTIVELY. FURTHERMORE, THEY SHOW THAT IF SUCH A SPINOR EXISTS, THEN $N=3$ OR $7$, GIVING THE MULTIPLICITIES $(M_1/2,M_-3/2)=(1,2)$ OR $(3,4)$. IN WHAT FOLLOWS WE EXAMINE EACH OF THE 3- AND 7-DIMENSIONAL CASES FROM OUR CLASSIFICATION AND DETERMINE WHETHER, WHEN EQUIPPED WITH THE ROUND METRIC, THEY ADMIT INVARIANT GENERALIZED KILLING SPINORS WITH EXACTLY TWO EIGENVALUES.
\BEGIN{ENUMERATE}[(I).]
\ITEM $G=\SO(4)$, $\SO(8)$, $\U(2)$, OR $\U(4)$. BY THEOREMS \REF{SOTHEOREM} AND \REF{UTHEOREM} THERE ARE NO INVARIANT SPINORS TO CONSIDER IN THESE CASES.
\ITEM $G=\SU(2)\CONG \SP(1)$. BY THEOREM \REF{DEFORMEDSASAKIANINVARIANTSPINORS} AND COROLLARY \REF{SUNWEAKPROP}, THE ROUND METRIC $G_{A,B}\RVERT_{A= B=\FRAC{1}{2}}$ ADMITS A PAIR OF INVARIANT KILLING SPINORS FOR THE CONSTANT $\FRAC{1}{2}$, BUT NO INVARIANT GENERALIZED KILLING SPINORS.
\ITEM $G=\SP(1)\SP(1)$. BY THEOREM \REF{EXPLICITSPINORSK=SP1} THERE ARE NO INVARIANT SPINORS TO CONSIDER IN THIS CASE.
\ITEM $G=\SP(1)\U(1)$. BY THEOREM \REF{EXPLICITSPINORSK=U1} THERE ARE NO INVARIANT SPINORS TO CONSIDER IN THIS CASE.
\ITEM $G=\SP(2)$. CONSIDERING EXAMPLE \REF{SP2SP1} WITH THE ROUND METRIC $G_{\ALPHA,\DELTA}\RVERT_{\ALPHA=\DELTA=1}$, ONE RECOVERS THE CANONICAL SPINOR DESCRIBED IN [5, 3], CONSISTENT ALSO WITH THEOREM 4.10 IN [40].
\ITEM $G=\SP(2)\SP(1)$. THE ROUND METRIC IN THIS CASE IS GIVEN BY $G_{A,B}\RVERT_{A=\FRAC{5}{24}, \ B=\FRAC{1}{24}}$. SUBSTITUTING THESE VALUES OF $A$ AND $B$ INTO THE ENDOMORPHISM $A$ FROM PROPOSITION \REF{SP1TWISTEDGKS} SHOWS THAT THE SPINOR $\PSI_0$ FROM THEOREM \REF{EXAMPLE_INVARIANT_SPINORS_SP_TWISTED_7SPHERE} IS AN INVARIANT GENERALIZED KILLING SPINOR WITH TWO DISTINCT EIGENVALUES, AND THE ASSOCIATED ENDOMORPHISM IS CONSISTENT WITH THEOREM 4.10 IN [40]; THIS THEOREM ALSO IMPLIES THE EXISTENCE OF A 3-SASAKIAN STRUCTURE WITH $\PSI_0$ AS THE CANONICAL SPINOR, HOWEVER THIS STRUCTURE CANNOT BE INVARIANT AS A CONSEQUENCE OF COROLLARY \REF{S7_SP2SP1_NO_INV_SASAKIAN_STRUCTURE}.
\ITEM $G=\SP(2)\U(1)$. THE ROUND METRIC IN THIS CASE IS GIVEN BY $G_{A,B,C}\RVERT_{A=\FRAC{5}{24}, \ B= \FRAC{1}{12}, \ C=\FRAC{1}{24}}$, AND BY COROLLARY \REF{SPNU1ALPHASASAKIANPARAMTERIZATION} THERE IS A COMPATIBLE INVARIANT SASAKIAN STRUCTURE. BY REMARK \REF{U1TWISTEDROUNDMETRIC}, THE SPINOR $\PSI_0$ FROM THEOREM \REF{U1TWISTEDGKS} IS AN INVARIANT GENERALIZED KILLING SPINOR WITH TWO DISTINCT EIGENVALUES, AND THE ASSOCIATED ENDOMORPHISM IS CONSISTENT WITH THEOREM 4.10 IN [40]; THIS THEOREM ALSO IMPLIES THE EXISTENCE OF A 3-SASAKIAN STRUCTURE WITH $\PSI_0$ AS THE CANONICAL SPINOR, HOWEVER THIS STRUCTURE CANNOT BE INVARIANT SINCE THE SPACE OF INVARIANT VECTORS IS 1-DIMENSIONAL.
\ITEM $G=\SPIN(7)$. IN THIS CASE, FOR ANY INVARIANT METRIC, THE 1-DIMENSIONAL SPACE OF INVARIANT SPINORS CONSISTS OF KILLING SPINORS.
\END{ENUMERATE}
%\SECTION{OPEN QUESTIONS}\TODO{REMOVE THIS SECTION}
%TO CONCLUDE, WE PRESENT SOME REMAINING OPEN QUESTIONS IN THIS AREA.
%	\ITEM LET $(M=G/H,G)$ BE A RIEMANNIAN HOMOGENEOUS MANIFOLD CARRYING AN INVARIANT KILLING SPINOR, $\PSI$. IF $\HAT{G}$ IS AN INVARIANT METRIC OBTAINED BY RESCALING $G$ SEPARATELY ON EACH ISOTROPY COMPONENT, FIND THE SPINORIAL EQUATION SATISFIED BY THE CORRESPONDING SPINOR $\HAT{\PSI}$ ON $(M,\HAT{G})$. NAIVELY ONE EXPECTS $\HAT{\PSI}$ TO BE A GENERALIZED KILLING SPINOR, HOWEVER PROPOSITION \REF{DEFORMATIONOFE1MINUS} SHOWS THAT THE SITUATION IS MORE COMPLICATED, EVEN IN THE CASE OF A RELATIVELY SIMPLE HOMOGENEOUS SPACE.
%	\ITEM FOR THE CASE $G=\SPIN(9)$, DOES THE INVARIANT SPINOR $\PSI$ INDUCE A GEOMETRIC STRUCTURE? IS THERE AN "OCTONIONIC CONTACT STRUCTURE" ON $S^{15}=\SPIN(9)/\SPIN(7)$ SUCH THAT $\PSI$ PLAYS A SIMILAR ROLE AS THE CANONICAL SPINOR $\PSI_0$ FROM [5] IN THE 3-SASAKIAN CASE?
\APPENDIX
\SECTION{AN EXPLICIT CONSTRUCTION OF THE LIE ALGEBRAS $\MATHFRAK{SU}(3)$, $\MATHFRAK{G}_2$, $\MATHFRAK{SPIN}(7)$, AND $\MATHFRAK{SPIN}(9)$}
THE EXCEPTIONAL CASES IN TABLE \REF{TAB:HOMOGENEOUSSPHERES} CAN BE GREATLY SIMPLIFIED BY CONSTRUCTING IN A UNIFIED MANNER THE LIE ALGEBRA INCLUSIONS $𝔰𝔲(3)⊂𝔤_2 ⊂𝔰𝔭𝔦𝔫(7) ⊂𝔰𝔭𝔦𝔫(9)$. THIS APPENDIX IS DEVOTED TO THE EXPOSITION OF THIS CONSTRUCTION, PARTS OF WHICH MAY BE FOUND IN CHAPER 4.4 OF [10]:
\BEGIN{LEMMA}
(BASED ON CHAPTER 4.4, LEMMA 15 IN [10]). THE LIE ALGEBRA $\MATHFRAK{G}_2$ (RESP. $\MATHFRAK{SU}(3)$) MAY BE REALIZED AS THE STABILIZER OF ONE (RESP. TWO) SPINORS IN THE REAL SPIN REPRESENTATION $\SIGMA_8:=\R^8$ OF $\MATHFRAK{SPIN}(7)$. EXPLICITLY, IF $\EPSILON_1,\DOTS\EPSILON_7$ IS THE STANDARD BASIS OF $\R^7$ AND $\PHI_1,\DOTS,\PHI_8$ IS THE STANDARD BASIS OF THE REAL SPINOR MODULE $\SIGMA_7=\R^8$, THEN $\MATHFRAK{G}_2$ AND $\MATHFRAK{SU}(3)$ ARE REALIZED INSIDE $\MATHFRAK{SPIN}(7)=\SPAN_{\R}\{ \EPSILON_I\EPSILON_J\}  $ VIA
\BEGIN{ALIGN*}
\MATHFRAK{G}_2& \CONG \TEXT{\EMPH{STAB}}_{\MATHFRAK{SPIN}(7)}\{\PHI_1 \} \CONG  \LEFT\{\SUM_{1\LEQ I<J \LEQ 7} \OMEGA_{I,J} \EPSILON_I\EPSILON_J\: \BEGIN{MULTLINED} \OMEGA_{1,2}+\OMEGA_{3,4} + \OMEGA_{5,6}=0, \\ -\OMEGA_{1,3}+\OMEGA_{2,4} -\OMEGA_{6,7}=0, \ -\OMEGA_{1,4}-\OMEGA_{2,3} -\OMEGA_{5,7} =0, \\ -\OMEGA_{1,6}-\OMEGA_{2,5} +\OMEGA_{3,7}=0, \ \OMEGA_{1,5}-\OMEGA_{2,6} -\OMEGA_{4,7}=0, \\ \OMEGA_{1,7}+\OMEGA_{3,6} +\OMEGA_{4,5} =0, \ \OMEGA_{2,7}+ \OMEGA_{3,5}-\OMEGA_{4,6}=0  \END{MULTLINED}     \RIGHT\} , \\
\MATHFRAK{SU}(3)&\CONG \TEXT{\EMPH{STAB}}_{\MATHFRAK{SPIN}(7)} \{\PHI_1,\PHI_2\} = \LEFT\{ \SUM_{1\LEQ I<J\LEQ 7} \OMEGA_{I,J}\EPSILON_I\EPSILON_J\: \BEGIN{MULTLINED} \OMEGA_{1,2}+\OMEGA_{3,4}+\OMEGA_{5,6}=0 \\ \OMEGA_{1,3}=\OMEGA_{2,4}, \ \OMEGA_{1,4}+\OMEGA_{2,3}=0, \  \OMEGA_{1,5}=\OMEGA_{2,6}, \\ \OMEGA_{1,6}+\OMEGA_{2,5}=0, \ \OMEGA_{3,5}=\OMEGA_{4,6}, \ \OMEGA_{3,6}+\OMEGA_{4,5}=0, \\ \OMEGA_{1,7}=\OMEGA_{2,7}=\DOTS=\OMEGA_{6,7}=0 \END{MULTLINED}    \RIGHT\}.
\END{ALIGN*}
\END{LEMMA}
\BEGIN{REMARK}
IN ORDER TO FIND BASES FOR THESE LIE ALGEBRAS WE USE THE EXPLICIT REALIZATION OF THE SPIN REPRESENTATION OBTAINED FROM THE FOLLOWING MATRICES:
\BEGIN{ALIGN*}
\RHO(\EPSILON_1) &:= E^{(8)}_{1, 8} + E^{(8)}_{2, 7} - E^{(8)}_{3, 6} - E^{(8)}_{4, 5}, \QUAD
\RHO(\EPSILON_2) := -E^{(8)}_{1, 7} + E^{(8)}_{2, 8} + E^{(8)}_{3, 5} -
E^{(8)}_{4, 6},\\
\RHO(\EPSILON_3) &:= -E^{(8)}_{1, 6} + E^{(8)}_{2, 5} - E^{(8)}_{3, 8} +
E^{(8)}_{4, 7},\QUAD
\RHO(\EPSILON_4) := -E^{(8)}_{1, 5} - E^{(8)}_{2, 6} - E^{(8)}_{3, 7} -
E^{(8)}_{4, 8},\\
\RHO(\EPSILON_5) &:= -E^{(8)}_{1, 3} - E^{(8)}_{2, 4} + E^{(8)}_{5, 7} +
E^{(8)}_{6, 8},\QUAD
\RHO(\EPSILON_6) := E^{(8)}_{1, 4} - E^{(8)}_{2, 3} - E^{(8)}_{5, 8} + E^{(8)}_{6, 7},\\
\RHO(\EPSILON_7) &:= E^{(8)}_{1, 2} - E^{(8)}_{3, 4} - E^{(8)}_{5, 6} + E^{(8)}_{7, 8}
\END{ALIGN*}
(SEE CHAPTER 4.4 IN [10]). BY SUBSTITUTING THESE INTO THE EQUATIONS OF THE PRECEDING LEMMA AND SUBSEQUENTLY ORTHOGONALIZING WITH RESPECT TO $B_0$, ONE OBTAINS THE FOLLOWING PROPOSITION.
\END{REMARK}
\BEGIN{PROPOSITION} \LABEL{ONBASISG2SU3}
A $B_0$-ORTHONORMAL BASIS FOR $\MATHFRAK{G}_2$ GIVEN BY
\BEGIN{ALIGN*}
\NU_1& := \FRAC{1}{4} (\RHO (\EPSILON_1) \RHO (\EPSILON_2)-\RHO (\EPSILON_5) \RHO (\EPSILON_6)),\QUAD \NU_2:= \FRAC{1}{4} (\RHO (\EPSILON_3) \RHO (\EPSILON_5)+\RHO (\EPSILON_4) \RHO (\EPSILON_6)), \\ \NU_3&:= \FRAC{1}{4} (\RHO (\EPSILON_3) \RHO (\EPSILON_6)-\RHO (\EPSILON_4) \RHO (\EPSILON_5)), \QUAD \NU_4:= \FRAC{1}{4} (\RHO (\EPSILON_1) \RHO (\EPSILON_3)+\RHO (\EPSILON_2) \RHO (\EPSILON_4)),\\ \NU_5&:= \FRAC{1}{4} (\RHO (\EPSILON_1) \RHO (\EPSILON_4)-\RHO (\EPSILON_2) \RHO (\EPSILON_3)), \QUAD \NU_6:= \FRAC{1}{4} (\RHO (\EPSILON_1) \RHO (\EPSILON_5)+\RHO (\EPSILON_2) \RHO (\EPSILON_6)),\\ \NU_7&:= \FRAC{1}{4} (\RHO (\EPSILON_1) \RHO (\EPSILON_6)-\RHO (\EPSILON_2) \RHO (\EPSILON_5)), \QUAD \NU_8:= -\FRAC{\RHO (\EPSILON_1) \RHO (\EPSILON_2)-2 \RHO (\EPSILON_3) \RHO (\EPSILON_4)+\RHO (\EPSILON_5) \RHO (\EPSILON_6)}{4 \SQRT{3}},\\ \NU_9&:= \FRAC{2 \RHO (\EPSILON_1) \RHO (\EPSILON_7)-\RHO (\EPSILON_3) \RHO (\EPSILON_6)-\RHO (\EPSILON_4) \RHO (\EPSILON_5)}{4 \SQRT{3}},\QUAD \NU_{10}:= \FRAC{2 \RHO (\EPSILON_2) \RHO (\EPSILON_7)-\RHO (\EPSILON_3) \RHO (\EPSILON_5)+\RHO (\EPSILON_4) \RHO (\EPSILON_6)}{4 \SQRT{3}}, \\ \NU_{11}&:= \FRAC{\RHO (\EPSILON_1) \RHO (\EPSILON_3)-\RHO (\EPSILON_2) \RHO (\EPSILON_4)-2 \RHO (\EPSILON_6) \RHO (\EPSILON_7)}{4 \SQRT{3}},\QUAD \NU_{12} :=\FRAC{\RHO (\EPSILON_1) \RHO (\EPSILON_4)+\RHO (\EPSILON_2) \RHO (\EPSILON_3)-2 \RHO (\EPSILON_5) \RHO (\EPSILON_7)}{4 \SQRT{3}},\\ \NU_{13}&:= \FRAC{\RHO (\EPSILON_1) \RHO (\EPSILON_5)-\RHO (\EPSILON_2) \RHO (\EPSILON_6)+2 \RHO (\EPSILON_4) \RHO (\EPSILON_7)}{4 \SQRT{3}},\QUAD \NU_{14}:= \FRAC{\RHO (\EPSILON_1) \RHO (\EPSILON_6)+\RHO (\EPSILON_2) \RHO (\EPSILON_5)+2 \RHO (\EPSILON_3) \RHO (\EPSILON_7)}{4 \SQRT{3}},
\END{ALIGN*}
WITH THE FIRST 8 ELEMENTS $\NU_1,\DOTS \NU_8$ FORMING A $B_0$-ORTHONORMAL BASIS FOR THE SUBALGEBRA $\MATHFRAK{SU}(3)$.
\END{PROPOSITION}
WE NOW WISH TO EXTEND THIS TO $B_0$-ORTHONORMAL BASES OF $𝔰𝔭𝔦𝔫(7)$ AND $𝔰𝔭𝔦𝔫(9)$. DENOTING BY $Ι  MAT_8()↪MAT_9()$ THE EMBEDDING AS THE LOWER RIGHT HAND $8×8$ BLOCK, ONE HAS,
\BEGIN{PROPOSITION} \LABEL{ONBASISSPIN7SPIN9}
THE BASIS $\{\NU_1,\DOTS,\NU_{14}\}$ EXTENDS TO A $B_0$-ORTHONORMAL BASIS OF $\MATHFRAK{SPIN}(7)$ GIVEN BY
AND A $B_0$-ORTHONORMAL BASIS OF $\MATHFRAK{SPIN}(9)$ GIVEN BY \LINEBREAK
\{\IOTA(\NU_1),\DOTS,\IOTA(\NU_{14}),\IOTA(\NU'_{15}),\DOTS,\IOTA(\NU'_{21}),\NU'_{22},\DOTS , \NU'_{36}\}
$, WHERE
\BEGIN{ALIGN*}
\NU'_{15}&:=
\FRAC{\RHO (\EPSILON_1) \RHO (\EPSILON_2)+\RHO (\EPSILON_3) \RHO (\EPSILON_4)+\RHO (\EPSILON_5) \RHO (\EPSILON_6)}{2 \SQRT{6}},\QUAD \NU'_{16}:= \FRAC{\RHO (\EPSILON_1) \RHO (\EPSILON_3)-\RHO (\EPSILON_2) \RHO (\EPSILON_4)+\RHO (\EPSILON_6) \RHO (\EPSILON_7)}{2 \SQRT{6}},\\ \NU'_{17}&:= \FRAC{\RHO (\EPSILON_1) \RHO (\EPSILON_4)+\RHO (\EPSILON_2) \RHO (\EPSILON_3)+\RHO (\EPSILON_5) \RHO (\EPSILON_7)}{2 \SQRT{6}},\QUAD \NU'_{18}:= \FRAC{-\RHO (\EPSILON_1) \RHO (\EPSILON_5)+\RHO (\EPSILON_2) \RHO (\EPSILON_6)+\RHO (\EPSILON_4) \RHO (\EPSILON_7)}{2 \SQRT{6}},\\ \NU'_{19}&:= -\FRAC{\RHO (\EPSILON_1) \RHO (\EPSILON_6)+\RHO (\EPSILON_2) \RHO (\EPSILON_5)-\RHO (\EPSILON_3) \RHO (\EPSILON_7)}{2 \SQRT{6}},\QUAD \NU'_{20}:=\QUAD \FRAC{\RHO (\EPSILON_1) \RHO (\EPSILON_7)+\RHO (\EPSILON_3) \RHO (\EPSILON_6)+\RHO (\EPSILON_4) \RHO (\EPSILON_5)}{2 \SQRT{6}}, \\ \NU'_{21}&:= -\FRAC{\RHO (\EPSILON_2) \RHO (\EPSILON_7)+\RHO (\EPSILON_3) \RHO (\EPSILON_5)-\RHO (\EPSILON_4) \RHO (\EPSILON_6)}{2 \SQRT{6}},
\END{ALIGN*}
AND {\SMALL
\BEGIN{ALIGN*}
\NU'_{22}&:= \SQRT{2}\LEFT(E_{2,3}^{(9)} -\SQRT{\FRAC{3}{2}} \  \IOTA(\NU'_{15}) \RIGHT), \QUAD \NU'_{23}:= \SQRT{2}\LEFT(E_{2,4}^{(9)} +\SQRT{\FRAC{3}{2}} \  \IOTA(\NU'_{16}) \RIGHT), \QUAD \NU'_{24}:=  \SQRT{2}\LEFT(E_{2,5}^{(9)} +\SQRT{\FRAC{3}{2}} \  \IOTA(\NU'_{17}) \RIGHT),\\
\NU'_{25}&:=  \SQRT{2}\LEFT(E_{2,6}^{(9)} -\SQRT{\FRAC{3}{2}} \  \IOTA(\NU'_{19}) \RIGHT),\QUAD \NU'_{26}:=  \SQRT{2}\LEFT(E_{2,7}^{(9)} +\SQRT{\FRAC{3}{2}} \ \IOTA( \NU'_{18}) \RIGHT), \QUAD \NU'_{27}:=  \SQRT{2}\LEFT(E_{2,8}^{(9)} -\SQRT{\FRAC{3}{2}} \ \IOTA( \NU'_{20}) \RIGHT), \\
\NU'_{28}&:=  \SQRT{2}\LEFT(E_{2,9}^{(9)} +\SQRT{\FRAC{3}{2}} \ \IOTA( \NU'_{21}) \RIGHT), \QUAD \NU'_{28+I}:=\FRAC{1}{\SQRT{2}} E^{(9)}_{1,1+I} \ \FORALL I=1,\DOTS,8.
\END{ALIGN*}
\END{PROPOSITION}
\NEWPAGE
\SECTION{INVARIANT DIFFERENTIAL FORMS ON $S^{15}=\SPIN(9)/\SPIN(7)$}\LABEL{APPENDIXB_FORMS}
HERE WE GIVE EXPLICIT FORMULAS, IN TERMS OF THE BASIS (\REF{S15ONB}), FOR THE DIFFERENTIAL FORMS ON $S^15=(9)/(7)$ DISCUSSED IN SECTION \REF{SPIN9SPIN7}:
\BEGIN{ALIGN*}
\OMEGA&:=   -E_{1,  8,  9}+E_{1,  10,  11}+E_{1,  12,  13}-E_{1,  14,  15}-E_{2,  8,  10}-E_{2,  9,  11}+E_{2,  12,  14}+E_{2,  13,  15} \\ &\QQUAD    -E_{3,  8,  11}+E_{3,  9,  10}+E_{3,  12,  15}-E_{3,  13,  14}-E_{4,  8,  12}-E_{4,  9,  13}-E_{4,  10,  14}-E_{4,  11,  15} \\ &\QQUAD   -E_{5,  8,  13}+E_{5,  9,  12}-E_{5,  10,  15}+E_{5,  11,  14}-E_{6,  8,  14}+E_{6,  9,  15}+E_{6,  10,  12}-E_{6,  11,  13} \\ &\QQUAD   -E_{7,  8,  15}-E_{7,  9,  14}+E_{7,  10,  13}+E_{7,  11,  12}     , \\
\PSI &:= E_{8,9,10,11} + E_{8,9,12,13} -E_{8,9,14,15} + E_{8,10,12,14} + E_{8,10,13,15} + E_{8,11,12,15} \\ &\QQUAD  -E_{8,11,13,14}   -E_{9,10,12,15}  + E_{9,10,13,14} + E_{9,11,12,14} + E_{9,11,13,15} \\ &\QQUAD  -E_{10,11,12,13} + E_{10,11,14,15} + E_{12,13,14,15} , \\
\SQRT{\FRAC{A}{2}} \ D\OMEGA &= E_{1,2,8,11}-E_{1,2,9,10}+E_{1,2,12,15}-E_{1,2,13,14}-E_{1,3,8,10}-E_{1,3,9,11}-E_{1,3,12,14}-E_{1,3,13,15}\\
&\QQUAD +E_{1,4,8,13}-E_{1,4,9,12}-E_{1,4,10,15}+E_{1,4,11,14}-E_{1,5,8,12}-E_{1,5,9,13}+E_{1,5,10,14}+E_{1,5,11,15}\\
&\QQUAD -E_{1,6,8,15}-E_{1,6,9,14}-E_{1,6,10,13}-E_{1,6,11,12}+E_{1,7,8,14}-E_{1,7,9,15}+E_{1,7,10,12}-E_{1,7,11,13}\\
&\QQUAD +E_{2,3,8,9}-E_{2,3,10,11}+E_{2,3,12,13}-E_{2,3,14,15}+E_{2,4,8,14}+E_{2,4,9,15}-E_{2,4,10,12}-E_{2,4,11,13}\\
&\QQUAD +E_{2,5,8,15}-E_{2,5,9,14}-E_{2,5,10,13}+E_{2,5,11,12}-E_{2,6,8,12}+E_{2,6,9,13}-E_{2,6,10,14}+E_{2,6,11,15}\\
&\QQUAD -E_{2,7,8,13}-E_{2,7,9,12}-E_{2,7,10,15}-E_{2,7,11,14}+E_{3,4,8,15}-E_{3,4,9,14}+E_{3,4,10,13}-E_{3,4,11,12}\\
&\QQUAD -E_{3,5,8,14}-E_{3,5,9,15}-E_{3,5,10,12}-E_{3,5,11,13}+E_{3,6,8,13}+E_{3,6,9,12}-E_{3,6,10,15}-E_{3,6,11,14}\\
&\QQUAD -E_{3,7,8,12}+E_{3,7,9,13}+E_{3,7,10,14}-E_{3,7,11,15}+E_{4,5,8,9}+E_{4,5,10,11}-E_{4,5,12,13}-E_{4,5,14,15}\\
&\QQUAD +E_{4,6,8,10}-E_{4,6,9,11}-E_{4,6,12,14}+E_{4,6,13,15}+E_{4,7,8,11}+E_{4,7,9,10}-E_{4,7,12,15}-E_{4,7,13,14}\\
&\QQUAD -E_{5,6,8,11}-E_{5,6,9,10}-E_{5,6,12,15}-E_{5,6,13,14}+E_{5,7,8,10}-E_{5,7,9,11}+E_{5,7,12,14}-E_{5,7,13,15}\\
&\QQUAD -E_{6,7,8,9}-E_{6,7,10,11}-E_{6,7,12,13}-E_{6,7,14,15}-\FRAC{3 A E_{8,9,10,11}}{2B}-\FRAC{3 A E_{8,9,12,13}}{2B}+\FRAC{3 A E_{8,9,14,15}}{2B}\\
&\QQUAD -\FRAC{3 A E_{8,10,12,14}}{2B}-\FRAC{3 A E_{8,10,13,15}}{2B}-\FRAC{3 A E_{8,11,12,15}}{2B}+\FRAC{3 A E_{8,11,13,14}}{2B}+\FRAC{3 A E_{9,10,12,15}}{2B}\\
&\QQUAD -\FRAC{3 A E_{9,10,13,14}}{2B} -\FRAC{3 A E_{9,11,12,14}}{2B}-\FRAC{3 A E_{9,11,13,15}}{2B}+\FRAC{3 A E_{10,11,12,13}}{2B}-\FRAC{3 A E_{10,11,14,15}}{2B}\\
&\QQUAD -\FRAC{3 A E_{12,13,14,15}}{2B}    \\
\SQRT{\FRAC{A}{2}} \ D\PSI&= E_{1,8,10,12,15}-E_{1,8,10,13,14}-E_{1,8,11,12,14}-E_{1,8,11,13,15}+E_{1,9,10,12,14}+E_{1,9,10,13,15}\\
&\QQUAD +E_{1,9,11,12,15}-E_{1,9,11,13,14}-E_{2,8,9,12,15}+E_{2,8,9,13,14}+E_{2,8,11,12,13}-E_{2,8,11,14,15}\\
&\QQUAD -E_{2,9,10,12,13}+E_{2,9,10,14,15}+E_{2,10,11,12,15}-E_{2,10,11,13,14}+E_{3,8,9,12,14}+E_{3,8,9,13,15}\\
&\QQUAD -E_{3,8,10,12,13}+E_{3,8,10,14,15}-E_{3,9,11,12,13}+E_{3,9,11,14,15}-E_{3,10,11,12,14}-E_{3,10,11,13,15}\\
&\QQUAD +E_{4,8,9,10,15}-E_{4,8,9,11,14}+E_{4,8,10,11,13}-E_{4,8,13,14,15}-E_{4,9,10,11,12}+E_{4,9,12,14,15}\\
&\QQUAD -E_{4,10,12,13,15}+E_{4,11,12,13,14}-E_{5,8,9,10,14}-E_{5,8,9,11,15}-E_{5,8,10,11,12}+E_{5,8,12,14,15}\\
&\QQUAD -E_{5,9,10,11,13}+E_{5,9,13,14,15}+E_{5,10,12,13,14}+E_{5,11,12,13,15}+E_{6,8,9,10,13}+E_{6,8,9,11,12}\\
&\QQUAD -E_{6,8,10,11,15}-E_{6,8,12,13,15}-E_{6,9,10,11,14}-E_{6,9,12,13,14}+E_{6,10,13,14,15}+E_{6,11,12,14,15}\\
&\QQUAD -E_{7,8,9,10,12}+E_{7,8,9,11,13}+E_{7,8,10,11,14}+E_{7,8,12,13,14}-E_{7,9,10,11,15}-E_{7,9,12,13,15}\\
&\QQUAD -E_{7,10,12,14,15}+E_{7,11,13,14,15}.
\END{ALIGN*}

WE ALSO RECORD IN TABLE \REF{TAB:ISOTROPYTYPESS15} THE \EMPH{ISOTROPY TYPES} OF THE FORMS $Ω, Ψ, DΩ, DΨ$ FROM SECTION \REF{SPIN9SPIN7}, I.E. THE NUMBER OF FACTORS FROM EACH ISOTROPY COMPONENT. THE ISOTROPY TYPES OF ALL OTHER INVARIANT FORMS IN TABLE \REF{TAB:INVARIANT_FORMS_S15} MAY BE EASILY DEDUCED FROM THESE.
\BEGIN{TABLE}[H!]
\CENTERING
\CAPTION{ISOTROPY TYPES OF INVARIANT FORMS ON $S^{15}=\SPIN(9)/\SPIN(7)$}
\BEGIN{TABULAR}{ |L||L| }
\HLINE
FORM & ISOTROPY TYPE \\
\HLINE
$\OMEGA $   &    $\MATHFRAK{M}_F\OTIMES \LAMBDA^2\MATHFRAK{M}_B$   \\
$\PSI$ & $\LAMBDA^4\MATHFRAK{M}_B$ \\
$D\OMEGA$ & $(\LAMBDA^2\MATHFRAK{M}_F \OTIMES \LAMBDA^2\MATHFRAK{M}_B) \OPLUS (\LAMBDA^4 \MATHFRAK{M}_B)$ \\
$D\PSI$ & $\MATHFRAK{M}_F\OTIMES \LAMBDA^4 \MATHFRAK{M}_B$ \\
\HLINE
\END{TABULAR}
\LABEL{TAB:ISOTROPYTYPESS15}
\END{TABLE}

\BIGSKIP
\BIGSKIP

\NEWCOMMAND{\ETALCHAR}[1]{$^{#1}$}
\BEGIN{THEBIBLIOGRAPHY}{BHM{\ETALCHAR{+}}15}

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\END{THEBIBLIOGRAPHY}

\BIBLIOGRAPHYSTYLE{ALPHA}

%[47] B. AMMANN, A. MOROIANU \& S. MOROIANU, \EMPH{THE CAUCHY PROBLEMS FOR EINSTEIN METRICS AND PARALLEL SPINORS}, COMM. MATH. PHYS. \TEXTBF{320} (2013) 173-198.

\END{DOCUMENT}