Let $\Phi[x, y]$ be the polynomial algebra over the field $\Phi$. Then $\Phi [x, y]$ is an alternative algebra too.
Let $A = \Phi [x,y]$ and $A_1 = \Phi[w, z]$, where $A_1$ is a vector space isomorphic to $A$.
In Jacobson's book, have the universal multiplicative envelope algebra of $A$ (I guess is called universal associative envelope in some texts), and can be calculate by
$$\mathfrak{U}(A) = \dfrac{T(A \oplus A_1)}{\mathfrak{R}}$$
where $T(A \oplus A_1) = \Phi \cdot 1 \oplus (A + A_1) \oplus [(A \oplus A_1) \otimes (A \oplus A_1)] \oplus \cdots$, the tensor algebra of $A \oplus A_1$. And $\mathfrak{R}$ is the ideal of $T(A \oplus A_1)$ generated by
$$a \otimes b_1 + a_1 \otimes b_1 - a_1b_1 - b_1 \otimes a = 0$$ $$a^2 - a \otimes a = 0 \Rightarrow a^2 = a \otimes a$$ $$a \otimes b + a_1 \otimes b - ba - b \otimes a_1 = 0$$ $$a_1^2 - a_1 \otimes a_1 = 0 \Rightarrow a_1^2 = a_1 \otimes a_1$$
In this case, $a$ and $b$ are monomials in the algebra $A$, $a_1$ and $b_1$ are monomials in the algebra $A_1$.
I do not know how to compute the $\mathfrak{U}(A)$, I would like a basis for $\mathfrak{U}(A)$.
If anyone can help me, I appreciate it.