Consider $(\mathbb{C}^2, \omega)$ where $\omega$ is a non-degenerate complex skew-symmetric bilinear form on $\mathbb{C}^2$. Let us write

$V = (\mathbb{C}^2, \omega)$

There are many spaces one can construct from $V$. For instance $Sym^n(V)$ is a vector space endowed with a non-degenerate complex bilinear form (respectively a non-degenerate complex skew-symmetric bilinear form) if n is even (respectively odd). Thus, we have realized $O(2k+1,\mathbb{C})$ and $Sp(2k,\mathbb{C})$ as groups of symmetry of a natural space constructed from $V$.

A more complicated example is $G_2$, which is the group of symmetries of $Sym^6(\mathbb{C}^2)$, endowed with a complex skew-symmetric trilinear form which can be defined using $\omega$ only. See for instance Theorem 1.1 (page 2) in https://arxiv.org/abs/1107.2813, though I am sure it goes back way before (not sure who is the first).

I am still struggling to get $O(2n,\mathbb{C})$ that way, as the group of symmetries of a natural space constructed from $V$, and I wonder whether the remaining exceptional Lie groups can be viewed this way.

I remark that there are more complicated spaces one can build from $V$, such as the kernel of the "symmetrization map":

$Sym^k(\mathbb{C}^2) \otimes Sym^l(\mathbb{C}^2) \to Sym^{k+l}(\mathbb{C}^2)$

or, the kernel of:

$Sym^k(\mathbb{C}^2) \otimes Sym^l(\mathbb{C}^2) \to Sym^{k-l}(\mathbb{C}^2)$

if $k \geq l$, the map being defined by contracting using the symplectic form $\omega$. (I believe that the right term for "natural space constructed from $V$" is a Schur functor, a term I have just met in one of Prof. Robert Bryant's answers to another post).

Edit 1: Allow me please to rephrase a little my question. Prof. @RobertBryant, while your answer is quite helpful, and I particularly thank you for the very nice description of $E_6$ that you have provided, yet what I would like to achieve, if possible, is different, though related.

Does there always exist, for any complex semisimple Lie algebra, a construction/definition/description of a corresponding compact real form, as a group of symmetries of $(W, \sigma)$, where $W$ is a complex vector space and $\sigma$ is a set of structures on $W$, under the conditions that $W$ can be constructed from the standard representation $V = \mathbb{C}^2$ of $SU(2)$ using (symmetrized, skew-symmetrized,...) tensor products of copies of $V$, and $\sigma$ can be constructed using the standard structures on $V$, namely the complex symplectic form $\omega \in \Lambda^2V^*$ and the quaternionic structure $j$ on $V$, which maps $(u,v)$ to $(-\bar{v}, \bar{u})$.

I already know that $Sp(m, \mathbb{R})$, $O(2m+1, \mathbb{R})$ and $G_2$ can be defined as groups of symmetries of some corresponding $(W, \sigma)$, where the latter is constructed from $(V, (\omega, j))$. We note also that $O(4, \mathbb{R})$ can also be defined as the group of symmetries of $(V \otimes V, (\sigma \otimes \sigma, j \otimes j))$. What about $O(2m, \mathbb{R})$ or $Spin(2m, \mathbb{R})$ and the other $4$ exceptional cases?

Edit 2: I would like to add that the complex $E_6$, using Prof. Robert Bryant's answer below, can also be defined using $SU(2)$ data. Indeed, consider first $(S^7(V), (S^7(\omega)))$. Note that $S^7(\omega)$ can play the role of what Prof. Bryant in his answer called $\omega$. Then one can form $\Lambda^2(S^7(V^*))$ and proceeds just as in his answer. While it is true that the $27$-dimensional complex space defined as the space of elements $\phi \in \Lambda^2(S^7(V^*))$ such that $\phi \wedge S^7(\omega)^3 = 0$, while it is not defined as a (symmetrized, skewsymmetrized,...) tensor product of copies of $V = \mathbb{C}^2$, yet the previous equation is defined only using $\omega$. I also allow that.

I have a question by the way. $S^7(j)$ is a quaternionic structure on $S^7(V)$, and $S^7(j) \wedge S^7(j)$ induces a real structure on $\Lambda^2(S^7(V^*))$. Is the group of symmetries of the previous $27$-dimensional complex space together with the cubic form and the real structure that I have just defined the compact real form $E_6$, or is it some other non-compact real form of $E_6^\mathbb{C}$? I am hoping it is the compact real form!