7 added 389 characters in body

This is more of an extended comment/enlargement of the original question/wild speculation than an answer. Due to my poor recollection of reading, a while back, Klein's Erlangen program and Franz Meyer's review on classical invariant theory, I thought Jose's Q 1 was an easy consequence of a 19th's century theorem which would say (disclaimer: what follows is meta-math):

Let $G$ be a group acting on some vector space $V$. Let $A$ denote a collection of tensors in some tensor powers of $V$ and its dual (such as $Q$ and $C$ above). Let $H$ be the subgroup of $G$ defined as the stabilizer of $A$. Suppose we know the first fundamental theorem for $G$: namely a description of invariants of $G$ using contractions of a finite set of elementary pieces. Then one can get the FFT of $H$ simply by adjoining the pieces in the collection $A$.'

Suppose for instance one looks at a polynomial $P({\mathbf{F}})$ where $\mathbf{F}$ is some generic $V$-tensor. Suppose $P(g{\mathbf{F}})=P(\mathbf{F})$ for all $g$ in $H$. Then the previous meta-theorem is the statement that there exists a polynomial $\Phi(\mathbf{F},\mathbf{A})$ involving this time a generic tensor or collection of tensors $\mathbf{A}$ of the same format as $A$, such that:

1) $\Phi$ is invariant by the full group $G$, i.e., $\Phi(g\mathbf{F},g\mathbf{A})=\Phi(\mathbf{F},\mathbf{A})$ for all $g$ in $G$.

2) the specialization of $\Phi(\mathbf{F},\mathbf{A})$ to $\mathbf{A}:=A$ is the original subgroup invariant $P(\mathbf{F})$.

If this were true then the FFT for $G=GL(V)$ would be the mother of all FFT's. The statement works for going from $GL(n)$ to $SL(n)$ and taking $A$ to be the epsilon Levi-civita tensor. For othogonal and symplectic groups one takes for $A$ a symmetric or antisymmetric form. But it works also if $G=SL(2)$ and $A=e_1=(1, 0)^T$ so that $H$ is the group of upper triangular matrices (I guess for $n>2$ one would need $A=e_1, e_1\wedge e_2,\ldots$). I thought the meta-theorem was mentioned in Franz Meyer's review, but in fact the only case he explicitly talks about is the last one. He calls that the study of "peninvariants" which is the same as seminvariants sources of covariants of binary forms etc.

Now here is a meta-meta-proof of the meta-theorem:

Let $\Phi(\mathbf{F},\mathbf{A}):=P(g\mathbf{F})$ where $g$ is some element of $G$ such that $\mathbf{A}=g^{-1}A$.

This is well defined since $g_1^{-1}A=g_2^{-1}A$ implies by definition of $H$ that $g_2 g_1^{-1}\in H$. Therefore $P(g_2\mathbf{F})=P((g_2 g_1^{-1})g_1\mathbf{F})= P(g_1\mathbf{F})$ by $H$-invariance of $P$.

This is a $G$-invariant because by construction, if $\mathbf{A}=g^{-1}A$ then $g'\mathbf{A}=(g g'^{-1})^{-1}A$ and so $\Phi(g'\mathbf{F},g'\mathbf{A})=P((gg'^{-1})g'\mathbf{F}) =P(g\mathbf{F})=\Phi(\mathbf{F},\mathbf{A})$.

Now of course there is a catch: one seems to need the $G$-orbit of $A$ to essentially (Zariski dense?) be all of the space where $\mathbf{A}$ belongs. This holds in the previous instances of the meta-theorem. Now when $A=C,Q$ as in Jose's problem, I am less sure. So I am not as convinced now by what I said in my comment above that Jose's $\Phi$ should be expressible in terms of $Q,C,\nu$ alone. Although, Bruce seems to be able to show this, so stay tuned.

In fact one does not seem to need the orbit of $A$ to fill everything, but one needs the existence of a $G$-invariant extension of the $\Phi$ I defined to the full space of $\mathbf{A}$. An idea to do this is to find some extension of $\Phi$ not necessarily invariant and then average it over $G$. For instance if $G=SL(n)$, one needs to take the Haar measure average over $SU(n)$. Usually if one does this the risk is to get zero, but this cannot happen here since the restriction of the average to $A$ should be the original $P$ which must be nonzero, otherwise the problem of expressing $P$ is moot. This sounds too good to be true, in particular in view of Nagata's counterexample for finite generation of rings of invariants. A specialist of algebraic groups would be better able to delineate the boundaries of what is solid math and what is pure fancy in the above arguments.

Since Jose's question 1 really is: is there a FFT for $F_4$? I briefly searched the literature and found this recent paper by Bruno Blind. It has some good references, in particular by Gerald Schwarz who solved this problem for $G_2$, and several papers by Iltyakov (who proves things about $F_4$ but uses notations and definitions I do not understand).

By the way, another paper by Schwarz in AIF is about binary cubics. This seems to be related to the instance of the meta-theorem where $G=SL(4)$ and $H$ is an imbedded $SL(2)$ using the 3rd symmetric power map. The $A$ in this case, I guess would be the twisted cubic curve. In general, one would need a hypersurface given by the Veronese imbedding (Hesse's transfer principle referred to in Klein's program). Classics studied the kind of things addressed in Schwarz's AIF article under the heading of "invariant types" see the book by Grace and Young Ch. XV and XVI.

The above setup also makes sense if $H$ is finite. I wonder if one can prove the fundamental theorem of symmetric polynomials (in a very complicated way) along these lines. I was toying with $A=x_1\cdots x_n$ or $x_1^p+\cdots+x_n^p$ for some well chosen power $p$ but one needs to get rid of a torus or roots of unity as well as solve the extension problem.

Update: I recently came across this article by Alexander Schrijver "Tensor subalgebras and first fundamental theorems in invariant theory". J. Algebra 319 (2008), no. 3, 1305–1319. It is related to what I said above since it deduces the FFT for classical groups from that of $GL(n)$.

6 corrected terminology

This is more of an extended comment/enlargement of the original question/wild speculation than an answer. Due to my poor recollection of reading, a while back, Klein's Erlangen program and Franz Meyer's review on classical invariant theory, I thought Jose's Q 1 was an easy consequence of a 19th's century theorem which would say (disclaimer: what follows is meta-math):

Let $G$ be a group acting on some vector space $V$. Let $A$ denote a collection of tensors in some tensor powers of $V$ and its dual (such as $Q$ and $C$ above). Let $H$ be the subgroup of $G$ defined as the stabilizer of $A$. Suppose we know the first fundamental theorem for $G$: namely a description of invariants of $G$ using contractions of a finite set of elementary pieces. Then one can get the FFT of $H$ simply by adjoining the pieces in the collection $A$.'

Suppose for instance one looks at a polynomial $P({\mathbf{F}})$ where $\mathbf{F}$ is some generic $V$-tensor. Suppose $P(g{\mathbf{F}})=P(\mathbf{F})$ for all $g$ in $H$. Then the previous meta-theorem is the statement that there exists a polynomial $\Phi(\mathbf{F},\mathbf{A})$ involving this time a generic tensor or collection of tensors $\mathbf{A}$ of the same format as $A$, such that:

1) $\Phi$ is invariant by the full group $G$, i.e., $\Phi(g\mathbf{F},g\mathbf{A})=\Phi(\mathbf{F},\mathbf{A})$ for all $g$ in $G$.

2) the specialization of $\Phi(\mathbf{F},\mathbf{A})$ to $\mathbf{A}:=A$ is the original subgroup invariant $P(\mathbf{F})$.

If this were true then the FFT for $G=GL(V)$ would be the mother of all FFT's. The statement works for going from $GL(n)$ to $SL(n)$ and taking $A$ to be the epsilon Levi-civita tensor. For othogonal and symplectic groups one takes for $A$ a symmetric or antisymmetric form. But it works also if $G=SL(2)$ and $A=e_1=(1, 0)^T$ so that $H$ is the group of upper triangular matrices (I guess for $n>2$ one would need $A=e_1, e_1\wedge e_2,\ldots$). I thought the meta-theorem was mentioned in Franz Meyer's review, but in fact the only case he explicitly talks about is the last one. He calls that the study of "peninvariants" which is the same as seminvariants sources of covariants of binary forms etc.

Now here is a meta-meta-proof of the meta-theorem:

Let $\Phi(\mathbf{F},\mathbf{A}):=P(g\mathbf{F})$ where $g$ is some element of $G$ such that $\mathbf{A}=g^{-1}A$.

This is well defined since $g_1^{-1}A=g_2^{-1}A$ implies by definition of $H$ that $g_2 g_1^{-1}\in H$. Therefore $P(g_2\mathbf{F})=P((g_2 g_1^{-1})g_1\mathbf{F})= P(g_1\mathbf{F})$ by $H$-invariance of $P$.

This is a $G$-invariant because by construction, if $\mathbf{A}=g^{-1}A$ then $g'\mathbf{A}=(g g'^{-1})^{-1}A$ and so $\Phi(g'\mathbf{F},g'\mathbf{A})=P((gg'^{-1})g'\mathbf{F}) =P(g\mathbf{F})=\Phi(\mathbf{F},\mathbf{A})$.

Now of course there is a catch: one seems to need the $G$-orbit of $A$ to essentially (Zariski dense?) be all of the space where $\mathbf{A}$ belongs. This holds in the previous instances of the meta-theorem. Now when $A=C,Q$ as in Jose's problem, I am less sure. So I am not as convinced now by what I said in my comment above that Jose's $\Phi$ should be expressible in terms of $Q,C,\nu$ alone. Although, Bruce seems to be able to show this, so stay tuned.

In fact one does not seem to need the orbit of $A$ to fill everything, but one needs the existence of a $G$-invariant extension of the $\Phi$ I defined to the full space of $\mathbf{A}$. An idea to do this is to find some extension of $\Phi$ not necessarily invariant and then average it over $G$. For instance if $G=SL(n)$, one needs to take the Haar measure average over $SU(n)$. Usually if one does this the risk is to get zero, but this cannot happen here since the restriction of the average to $A$ should be the original $P$ which must be nonzero, otherwise the problem of expressing $P$ is moot. This sounds too good to be true, in particular in view of Nagata's counterexample for finite generation of rings of invariants. A specialist of algebraic groups would be better able to delineate the boundaries of what is solid math and what is pure fancy in the above arguments.

Since Jose's question 1 really is: is there a FFT for $F_4$? I briefly searched the literature and found this recent paper by Bruno Blind. It has some good references, in particular by Gerald Schwarz who solved this problem for $G_2$, and several papers by Iltyakov (who proves things about $F_4$ but uses notations and definitions I do not understand).

By the way, another paper by Schwarz in AIF is about binary cubics. This seems to be related to the instance of the meta-theorem where $G=SL(4)$ and $H$ is an imbedded $SL(2)$ using the 3rd symmetric power map. The $A$ in this case, I guess would be the twisted cubic curve. In general, one would need a hypersurface given by the Veronese imbedding (Hesse's transfer principle referred to in Klein's program). Classics studied the kind of things addressed in Schwarz's AIF article under the heading of "invariant types" see the book by Grace and Young Ch. XV and XVI.

The above setup also makes sense if $H$ is finite. I wonder if one can prove Vieta's the fundamental theorem about of symmetric functions polynomials (in a very complicated way) along these lines. I was toying with $A=x_1\cdots x_n$ or $x_1^p+\cdots+x_n^p$ for some well chosen power $p$ but one needs to get rid of a torus or roots of unity as well as solve the extension problem.

5 added 698 characters in body
An idea to do this is to find some extension of $\Phi$ not necessarily invariant and thenaverage it over $G$. For instance if $G=SL(n)$, one needs to take the Haar measure average over $SU(n)$. Usually if one does this the risk is to get zero, but this cannot happen heresince the restriction of the average to $A$ should be the original $P$ which must be nonzero, otherwise the problem of expressing $P$ is moot. This sounds too good to be true, in particular in view of Nagata's counterexample for finite generation of rings of invariants. A specialist of algebraic groups would be better able to delineate the boundaries of what is solid math and what is pure fancy in the above arguments.