Is the triple product in a Freudenthal triple system fully symmetric?

You say that the 4-linear form $Q$ is obtained by 'linearization' of the quartic form $q$. (I can't stand giving them both the same name.) There will be many 4-linear forms $Q$ related to $q$ by

$$ Q(x,x,x,x) = q(x) $$

but, except perhaps in characteristic 2 or 3, there is just one that is totally symmetric, meaning that

$$ Q(x_{\sigma(1)}, \dots, x_{\sigma(4)}) = Q(x_1, \dots, x_4)$$

for all $\sigma \in S_4$. In this case there's a formula for $Q$ in terms of $q$, and people usually say it's obtained from $q$ by 'polarization'.

I mention this because you don't say in detail how $Q$ is obtained from $q$. If it's obtained by polarization, it will be totally symmetric - and then the triple product will be symmetric in all 3 arguments, solving your mystery.

I figured out the precise relationship between the 'Class of Ternary Algebras' of Faulkner and the 'Freudenthal Triple Systems' of Ferrar/Helenius and I will write it down here for the benefit of future readers (mostly my future self).

Following Helenius (as in the OP) I write $\langle . , . \rangle$ for the skew-symmetric biliear form. Following John Baez above I write $q$ for the quartic form and $Q$ for the 4-linear form. Following Faulkner I write $\langle . , . , . \rangle$ for the triple product. In all cases I use the subscript $f$ for Faulkners definitions and $h$ for Ferrar's definition (as I mostly learned his definition from Helenius). With these notational conventions we have:

$$\langle x, y \rangle_f = \langle y, x \rangle_h = - \langle x, y \rangle_h$$ $$Q_h(x_1, x_2, x_3, x_4) = \frac{1}{4!} \sum_{\pi \in S_4} Q_f(x_{\pi(1)}, x_{\pi(2)}, x_{\pi(3)}, x_{\pi(4)})$$ from which it follows that $$q_h(x) = Q_h(x, x, x, x) = Q_f(x, x, x, x) = q_f(x).$$ The relation between the triple products can be deduced from $$Q_f(x, y, z, w) = \langle \langle x, y ,z \rangle_f, w \rangle_f$$ $$Q_h(x, y, z, w) = \langle w, \langle x, y ,z \rangle_h \rangle_h = \langle \langle x, y ,z \rangle_h, w \rangle_f$$

This is not perfect. It allows us to deduce $Q_h$ from $Q_f$ but not the other way around and obtaining an explicit description of $\langle . , . , . \rangle_h$ in terms of $\langle . , . , . \rangle_f$ is messy. However, the article of Helenius provides a great way of clearing this up.

Faulkner starts with his version of the triple system and use it to create a Lie algebra out of it, similar constructions of (the same) Lie algebra from Ferrar's version of the triple system have been given elsewhere. Helenius turns this viewpoint upside down: he starts with the Lie algebra ($\mathfrak{g}$ in his notation) identifies a special subspace $\mathfrak{g}_1$ on which he can define $\langle . , . \rangle_h, \langle . , . , . \rangle_h$ and $Q_h$ in terms of the Lie algebra structure and then shows they satisfy the Freudenthal triple system axioms. I will give here the analogous presentation of $\langle . , . \rangle_f, \langle . , . , . \rangle_f$ and $Q_f$. We may assume that the Lie-algebra $\mathfrak{g}$ is simple, apparently this is equivalent to the non-degeneracy of $\langle . , . \rangle$.

Helenius starts with picking a long root $\rho$ and defines the space $\mathfrak{g}_1$ as the sum of all the root spaces $\mathfrak{g}_\alpha$ for roots $\alpha$ satisfying $2 \frac{(\alpha, \rho)}{(\rho, \rho)} = 1$. (I try to avoid introducing another use of $\langle . , . \rangle$ here.) More generally he defines a grading of $\mathfrak{g}$ where $\mathfrak{g}_k$ is the span of the rootspaces $\mathfrak{g}_\beta$ where $2 \frac{(\beta, \rho)}{(\rho, \rho)} = k$.

We fix a generator $x_{\rho}$ of $\mathfrak{g}_\rho$ and define $x_{-\rho} \in \mathfrak{g}_{-\rho}$ in such a way that $\{x_\rho, x_{-\rho}, [x_\rho, x_{-\rho}]\}$ is a standard $\mathfrak{sl}_2$-triple.

Now we have for all $x, y, z, w \in \mathfrak{g}_1$ that

$$[x, y] = \langle y, x \rangle_f x_\rho$$ $$\langle x, y, z \rangle_f = [x, [y, [z, x_{-\rho}]]] = (ad(x)\circ ad(y) \circ ad(z)) (x_{-\rho})$$ $$Q_f(x, y, z, w) = [w, [x, [y, [z, x_{-\rho}]]]] = (ad(w) \circ ad(x)\circ ad(y) \circ ad(z)) (x_{-\rho})$$

Comparing to the equalities $$[x, y] = \langle x, y \rangle_h x_\rho$$ $$Q_h(x_1, x_2, x_3, x_4) = \frac{1}{4!} \sum_{\pi \in S_4} (ad(x_{\pi(1)}) \circ ad(x_{\pi(2)}) \circ ad(x_{\pi(3)}) \circ ad(x_{\pi(4)})) (x_{-\rho})$$ from Helenius, chapter 3, yields the above relations.

The equation $\langle x, y, z \rangle_f = [x, [y, [z, x_{-\rho}]]]$ is given explicitely in Faulkner as part of the proof of Lemma 3. Verifying that the above descriptions of $\langle . , . \rangle_f, \langle . , . , . \rangle_f$ and $Q_f$ when taken as the definitions satisfy Faulkner's axioms $T1 - T4$ is mostly a long list of application of the Jacobi-idenity together with an occasional use of the fact that the space $\mathfrak{g}_k$ is zero for $k < - 2$ and $k > 2$.