The answer on math.SE points to a construction that produces an algebra of the desired type. (But maybe not all of them?) Here is a different kind of answer: There are uncountably many isomorphism classes of unital division algebras of dimension $n = 4$ or 8. Of course, only a single one of these is $\mathbb{H}$ or $\mathbb{O}$, so really there are a profusion of other ones out there.
I’ll point out the exact spot in the argument where the hypothesis that $n = 4$ or 8 gets used.
Consider the space of possible algebra structures on $\mathbb{R}^n$. What does it look like? We need to specify the multiplication, which is an element of the vector space $\mathbb{R}^n \otimes (\mathbb{R}^n)^* \otimes (\mathbb{R}^n)^*$, so that is the space of all algebra structures. But we also want the algebra to be unital. Given a unital multiplication on $\mathbb{R}^n$, we could do a change of basis to make the identity element $e$ the first basis vector. A multiplication with that property is an element of the vector space
$$
\mathcal{A} := \mathbb{R}^n \otimes (\mathbb{R}^{n-1})^* \otimes (\mathbb{R}^{n-1})^*.
$$
In summary, every element of $\mathcal{A}$ specifies a unital algebra structure on $\mathbb{R}^n$ with $e$ as identity, and every such algebra arises in this way.
Note that
$$
\dim \mathcal{A} = n(n-1)^2.
$$
Inside this vector space, define $\mathcal{D}$ to be the subset consisting of multiplications that define division algebras. There is a geometric argument in section 5 of
that shows that $\mathcal{D}$ is an open subset of $\mathcal{A}$. Now, since $n = 4$ or 8, $\mathcal{D}$ is not empty (because it contains $\mathbb{H}$ or $\mathbb{O}$), therefore we have:
$$
\dim \mathcal{D} = \dim \mathcal{A}.
$$
Next, let $P$ denote the subgroup of $GL_n(\mathbb{R})$ that fixes the first basis vector $e$. We have
$$
dim P = n(n-1),
$$
because it is the collection of invertible matrices whose first column is the transpose of $(1, 0, \ldots, 0)$.
Let $P$ act on $\mathcal{A}$ as $GL_n(\mathbb{R})$ naturally does, i.e., for $p \in P$, $x \in \mathbb{R}^n$, and $a, b \in (\mathbb{R}^{n-1})^*$:
$$
p(x \otimes a \otimes b) = px \otimes ap^{-1} \otimes bp^{-1}.
$$
Then $P$-orbits in $\mathcal{A}$ are exactly isomorphism classes of unital multiplications on $\mathbb{R}^n$ where the first basis vector is the identity.
Finally, let’s look at the orbit space $\mathcal{D} / P$. It parameterizes the isomorphism classes of the desired kinds of algebras. It has dimension
$$
\dim \mathcal{D}/P \ge \dim \mathcal{D} - \dim P = n(n-1)^2 - n(n-1) = n(n-1)(n-2).
$$
Since $n > 2$, this dimension is positive, so $\mathcal{D} / P$ has uncountably many points.
Definitions for reference
Because there are many similar-sounding results in the area based on different hypotheses, here are a few definitions that are used in this answer.
- A (real) algebra is a real vector space $A$ together with an $\mathbb{R}$-bilinear map $A \times A \to A$ that we call multiplication.
- The algebra is unital if there exists an element $e \in A$ such that $ea = ae = a$ for all $a \in A.$
- It is a division algebra if and only if, for every nonzero $x \in A$, the left multiplication map $L_x: A \to A$ defined by $L_x(y) = xy$ is invertible. (Usually the definition includes also that right multiplication by $x$ is also an invertible, but that is redundant since we are assuming that $A$ is finite-dimensional. Indeed, if right multiplication by a nonzero $x$ is not invertible, then there is a nonzero $y$ such that $yx = 0$, in which case left multiplication by $y$ is not invertible.)
Remarks
- This argument is essentially straight from Petersson’s paper, but with a few details changed to apply it to the question being posed. Petersson proves that there are uncountably many division algebras of dimension 4 or 8 up to isotopy (a weaker/coarser equivalence relation than isomorphism).
- I have heard studying the topology of $\mathcal{A}$ and similar spaces referred to by the charming name “algebraic geography”.
- If anyone who comes across this question is interested in the 2-dimensional case, my favorite reference for that case is
