Let $A$ be a commutative algebra of finite dimension over a field $F$.
Then $A\otimes A$ is also a commutative algebra. Clearly if $u_1,u_2$ are units in $A$, then $u_1\otimes u_2$ is a unit in $A\otimes A$; but it seems that not all units in $A\otimes A$ are of such form.
Any simple example of this? (I would like to consider tensor product of only two copies of $A$, and $A$ to be of smaller dimension $>1$ over given field.
$A=\mathbb Q[i]$ is an algebra of finite dimension over $\mathbb Q$. All elements of $A$ are units(?), but $i\otimes 1 + 1\otimes i \in A\otimes_{\mathbb Q} A$ is not of the form unit$\otimes$unit.
However, $$ (i\otimes 1 + 1\otimes i)^2 = (-1)\otimes 1 + 1\otimes(-1) + 2(i\otimes i) =2 (i\otimes i) $$ and $(i\otimes i)^4 = 1\otimes 1$ is a unit, so $i\otimes i$ is aswell.