2 fixed grammar

As I understand the question, the OP would be happy to see a description of the lowest-dimension fundamental representation of $F_4$ (and perhaps $E_6$, $E_7$, $E_8$), and is happy with the description of $G_2$ acting on the space of trace-zero octonions.

Yes - people have "bothered to write down" representations of $F_4$ and the other exceptional groups in this spirit. Chevalley, Schafer, Albert, Jacobson, Freudenthal, and Tits are the names that come to mind first.

$F_4$ acts naturally as automorphisms of the 27-dimensional exceptional Jordan algebra $J_{3,O}$ -- this is the Jordan algebra of 3 by 3 Hermitian matrices with entries in the octonions (which octonion algebra you use affects depends on or determines the form of $F_4$). Since $F_4$ acts as algebra automorphisms, preserving it preserves the unit element of this algebra, and preserving preserves the trace form as a result. It follows that $F_4$ acts on the 26-dimensional trace-zero subspace of the 27-dimensional algebra. This is quite close, in spirit, to the example of $G_2$ acting on trace-zero octonions.

Also, $F_4$ acts on this 26-dimensional space, preserving the nondegenerate symmetric trace form: $$(X,Y) \rightarrow Tr(X \cdot Y).$$

$E_6$ also acts on the 27-dimensional Jordan algebra above, but not as algebra automorphisms. Instead, $E_6$ can be viewed as the linear automorphisms of this 27-dimensional space that preserve the cubic norm form (the "determinant" of a 3 by 3 Hermitian octonionic matrix). I believe this goes back to Chevalley and Schafer about 60 years ago.

$E_7$ acts naturally on a 56-dimensional space, studied by Freudenthal. This is the space of two-by-two matrices, with diagonal entries in the base field, and off-diagonal entries in the exceptional Jordan algebra mentioned above: $2+27+27 = 56$. $E_7$ can be viewed as the group of linear automorphisms of this 56-dimensional space preserving a quartic form, I believe.

The smallest irreducible representation of $E_8$ is the adjoint representation of $E_8$ on its own Lie algebra -- so you have to construct $E_8$ to represent it, in a sense.

A nice recent survey of related topics, and a source of other references is Baez's survey on the octonions.

1

As I understand the question, the OP would be happy to see a description of the lowest-dimension fundamental representation of $F_4$ (and perhaps $E_6$, $E_7$, $E_8$), and is happy with the description of $G_2$ acting on the space of trace-zero octonions.

Yes - people have "bothered to write down" representations of $F_4$ and the other exceptional groups in this spirit. Chevalley, Schafer, Albert, Jacobson, Freudenthal, and Tits are the names that come to mind first.

$F_4$ acts naturally as automorphisms of the 27-dimensional exceptional Jordan algebra $J_{3,O}$ -- this is the Jordan algebra of 3 by 3 Hermitian matrices with entries in the octonions (which octonion algebra you use affects the form of $F_4$). Since $F_4$ acts as algebra automorphisms, preserving the unit element of this algebra, and preserving the trace form as a result. It follows that $F_4$ acts on the 26-dimensional trace-zero subspace of the 27-dimensional algebra. This is quite close, in spirit, to the example of $G_2$ acting on trace-zero octonions.

Also, $F_4$ acts on this 26-dimensional space, preserving the nondegenerate symmetric trace form: $$(X,Y) \rightarrow Tr(X \cdot Y).$$

$E_6$ also acts on the 27-dimensional Jordan algebra above, but not as algebra automorphisms. Instead, $E_6$ can be viewed as the linear automorphisms of this 27-dimensional space that preserve the cubic norm form (the "determinant" of a 3 by 3 Hermitian octonionic matrix). I believe this goes back to Chevalley and Schafer about 60 years ago.

$E_7$ acts naturally on a 56-dimensional space, studied by Freudenthal. This is the space of two-by-two matrices, with diagonal entries in the base field, and off-diagonal entries in the exceptional Jordan algebra mentioned above: $2+27+27 = 56$. $E_7$ can be viewed as the group of linear automorphisms of this 56-dimensional space preserving a quartic form, I believe.

The smallest irreducible representation of $E_8$ is the adjoint representation of $E_8$ on its own Lie algebra -- so you have to construct $E_8$ to represent it, in a sense.

A nice recent survey of related topics, and a source of other references is Baez's survey on the octonions.