fa.functional analysis - What's the nearest algebraic theory to inner product spaces?

Certainly you can get fairly close with your ternary operation $T(x,y,z) = langle x,y rangle z$. You can impose conditions on $T$ so that it comes from a bilinear form that takes values in a commutative ring of extended scalars acting on the vector space $V$. This is not entirely a bad thing; you could instead start with an abelian group $A$ and let $T$ induce both the bilinear form and the ground ring. To stick close to your original question, let's suppose that $V$ is a vector space over a fixed field $k$, and that the elements of $k$ are written into the algebraic theory. Then still the scalars could extend further, but you can write axioms to make sure that that is all that happens.

In detail, you can first suppose that $T(x,y,z)$ is trilinear and that $T(x,y,z) = T(y,x,z)$. Then $T$ can already be read as a bilinear form that takes values in operators acting on $V$. We can use the shorthand $U(z) = T(a,b,z)$, with $U$ an implicit function of $a$ and $b$, and see what conditions can be further imposed. You can impose the axiom:
$$T(a,b,T(c,d,x)) = T(c,d,T(a,b,x)),$$
which says that the different values of $U$ all commute, and thus generate a commutative algebra $R$. You can also impose the relation:
$$T(T(a,b,x),y,z) = T(a,b,T(x,y,z)),$$
in other words $T(U(x),y,z) = U(T(x,y,z))$. This relation says that $T$, as an operator-valued inner product, is $U$-linear.

With these relations, every word $W$ in $T$ collapses like this, after permuting inputs:
$$W(x_0,x_1,ldots,x_{2n}) = langle x_1, x_2 rangle cdots langle x_{2n-1}, x_{2n} rangle x_0,$$
where the product is interpreted over $R$ rather than over $k$. (The number of inputs must be even because $T$ is ternary.) This sort of collapse is the most that you can expect from any $(n,1)$-ary operation formed from a bilinear form. I think that that proves that with this approach, you can't do better than inner products with extension of scalars.

For all I know, it is possible that you could hard-code Euclidean geometry in some more subtle way using inequalities and names of elements in $mathbb{C}$ or $mathbb{R}$ in addition to using multilinear algebra. I do not know how to do that, though.