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This is an edited version of the original question taking into account the comments below by Bruce. The original formulation was imprecise.

It is well-known that $\mathfrak{g}$ leaves invariant a quadratic form $Q \in \operatorname{Sym}^2 V$ and a cubic form $C \in \operatorname{Sym}^3V$ on $V$. Indeed, $\mathfrak{g}$ can be characterised as the Lie subalgebra of $\mathfrak{sl}(V)$ which leaves invariant $Q$ and $C$. Since $V$ is irreducible, $Q$ is non-degenerate and we may use it to identify $V$ with $V^*$ as $\mathfrak{g}$ modules.

It seems to be part of the group theoretical folklore in the Physics literature (starting possibly with this paper) that any $\mathfrak{g}$-invariant tensor on $V$ --- that is, any $\mathfrak{g}$-invariant element of the tensor algebra of in $V$ \bigoplus_{n\geq 0} V^{\otimes n}$--- can be constructed out of$Q$(and its inverse),$C$and a nonzero "volume element"$\nu \in \Lambda^{26}V$. A Lambda^{26}V$ via products in the tensor algebra and contractions.

For example, we can construct six invariant tensors out of $Q$ and $C$ in degree $4$ Q_{ab}Q_{cd} \qquad Q_{ac}Q_{bd} \qquad Q_{ad}Q_{bc} \qquad C_{abe}C_{cdf} Q^{ef} \qquad C_{ace}C_{bdf} Q^{ef} \qquad C_{ade}C_{bcf} Q^{ef}which satisfy a linear relation, since there is only a 5-dimensional space of such tensors.

Now, a quick calculation in LiE , however, shows reveals that there is a $\mathfrak{g}$-invariant tensor $\Phi \in \Lambda^9 V$

which I would have a hard time constructing cannot be constructed out of $Q$, $C$ and $\nu$.\nu$in the aforementioned way. • Do • Can every invariant tensor be constructed out of$Q,C,\Phi,\nu$form a complete generating set for the Q$ (and its inverse) $F_4$-invariants in C$,$\bigotimes V$\nu$ and $\Phi$ by products and contractions?

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Let $\mathfrak{g}$ denote a complex simple Lie algebra of type $F_4$. Its smallest nontrivial irreducible representation is 26-dimensional. Let's call it $V$. This question is about the invariants of $\mathfrak{g}$ in this representation.

It is well-known that $\mathfrak{g}$ leaves invariant a quadratic form $Q \in \operatorname{Sym}^2 V$ and a cubic form $C \in \operatorname{Sym}^3V$ on $V$. Indeed, $\mathfrak{g}$ can be characterised as the Lie subalgebra of $\mathfrak{sl}(V)$ which leaves invariant $Q$ and $C$.

It seems to be part of the group theoretical folklore in the Physics literature (starting possibly with this paper) that any $\mathfrak{g}$-invariant tensor on $V$ --- that is, any $\mathfrak{g}$-invariant element of the tensor algebra of $V$ --- can be constructed out of $Q$, Q$(and its inverse),$C$and a nonzero "volume element"$\nu \in \Lambda^{26}V$. A quick calculation in LiE, however, shows that there is a$\mathfrak{g}$-invariant tensor$\Phi \in \Lambda^9 V$> alt_tensor(9,[0,0,0,1],F4)|[0,0,0,0] 1 > which I would have a hard time constructing out of$Q$,$C$and$\nu$. One possible way to understand$\Phi$is to think in terms of the$\mathfrak{so}(9)$subalgebra of$\mathfrak{g}$. Under$\mathfrak{so}(9)$,$V$breaks up as a direct sum of the trivial ($\Lambda^0$), vector ($\Lambda^1$) and spinor ($\Delta$) irreducible representations: $$V = \Lambda^0 \oplus \Lambda^1 \oplus \Delta$$ There are precisely two$\mathfrak{so}(9)$-invariants in$\Lambda^9 V$: one is the volume form on$\Lambda^1$and the other is the "volume" form on$\Lambda^0$wedged with the$\mathfrak{so}(9)$-invariant 8-form on$\Delta$. Notice that$(\mathfrak{so}(9),\Delta)$is the holonomy representation for the Cayley plane$F_4/\operatorname{Spin}(9)$, which is well-known to have a parallel self-dual$8$-form. Then$\Phi$is some linear combination of these two$\mathfrak{so}(9)$-invariants, which I have yet to work out. Questions I have two questions and, as usual, I would be very grateful for any pointers to the relevant literature: 1. Do$Q,C,\Phi,\nu$form a complete generating set for the$F_4$-invariants in$\bigotimes V$? 2. Is there a more convenient (for calculations) description of$\Phi$? In particular, I would like to know about the relation of the form$\Phi \otimes \Phi = \cdots$. Thank you in advance. 3 deleted 381 characters in body Let$\mathfrak{g}$denote a complex simple Lie algebra of type$F_4$. Its smallest nontrivial irreducible representation is 26-dimensional. Let's call it$V$. This question is about the invariants of$\mathfrak{g}$in this representation. It is well-known that$\mathfrak{g}$leaves invariant a quadratic form$Q \in \operatorname{Sym}^2 V$and a cubic form$C \in \operatorname{Sym}^3V$on$V$. Indeed,$\mathfrak{g}$can be characterised as the Lie subalgebra of$\mathfrak{sl}(V)$which leaves invariant$Q$and$C$. It seems to be part of the group theoretical folklore in the Physics literature (starting possibly with this paper) that any$\mathfrak{g}$-invariant tensor on$V$--- that is, any$\mathfrak{g}$-invariant element of the tensor algebra of$V$--- can be constructed out of$Q$,$C$and a nonzero "volume element"$\nu \in \Lambda^{26}V$. A quick calculation in LiE, however, shows that there is a$\mathfrak{g}$-invariant tensor$\Phi \in \Lambda^9 V$> alt_tensor(9,[0,0,0,1],F4) 1X[0,0,0,0] +1X[0,0,0,1] +2X[0,0,0,2] +2X[0,0,0,3] +2X[0,0,0,4] + 1X[0,0,0,5] +1X[0,0,1,0] +2X[0,0,1,1] +2X[0,0,1,2] +1X[0,0,1,3] + 3X[0,0,2,0] +1X[0,0,2,1] +1X[0,1,0,0] +2X[0,1,0,1] +1X[0,1,0,2] + 2X[0,1,1,0] +1X[0,2,0,0] +1X[1,0,0,1] +1X[1,0,0,2] +1X[1,0,0,3] + 3X[1,0,1,0] +3X[1,0,1,1] +1X[1,0,1,2] +1X[1,1,0,0] +1X[1,1,0,1] + 1X[2,0,0,0alt_tensor(9,[0,0,0,1],F4)|[0,0,0,0] +1X[2,0,0,1] +1X[2,0,0,2] 1 > which I would have a hard time constructing out of$Q$,$C$and$\nu$. One possible way to understand$\Phi$is to think in terms of the$\mathfrak{so}(9)$subalgebra of$\mathfrak{g}$. Under$\mathfrak{so}(9)$,$V$breaks up as a direct sum of the trivial ($\Lambda^0$), vector ($\Lambda^1$) and spinor ($\Delta$) irreducible representations: $$V = \Lambda^0 \oplus \Lambda^1 \oplus \Delta$$ There are precisely two$\mathfrak{so}(9)$-invariants in$\Lambda^9 V$: one is the volume form on$\Lambda^1$and the other is the "volume" form on$\Lambda^0$wedged with the$\mathfrak{so}(9)$-invariant 8-form on$\Delta$. Notice that$(\mathfrak{so}(9),\Delta)$is the holonomy representation for the Cayley plane$F_4/\operatorname{Spin}(9)$, which is well-known to have a parallel self-dual$8$-form. Then$\Phi$is some linear combination of these two$\mathfrak{so}(9)$-invariants, which I have yet to work out. Questions I have two questions and, as usual, I would be very grateful for any pointers to the relevant literature: 1. Do$Q,C,\Phi,\nu$form a complete generating set for the$F_4$-invariants in$\bigotimes V$? 2. Is there a more convenient (for calculations) description of$\Phi$? In particular, I would like to know about the relation of the form$\Phi \otimes \Phi = \cdots\$.