I don't understand the following notation:
$$V_F := V \otimes_k F$$
First of all, I know that that the product is a bilinear operation, i.e. $A \otimes A \to A$, between elements of the vector space $A$ in the algebra, but $F$ is a field, isn't it? $K$ is indeed a subfield of the bigger field $F$ with the operation restricted like in the classical example of $\mathbb R$ and $\mathbb C$. I've found a similar question and answer for vector spaces, and it explains that
$V_K$ is spanned by symbols of the form $a \otimes v$
but there it is noted that
these rules are not enough to combine every sum into an element of the form $a \otimes v$.
Therefore here, in the more complicated case of an algebra instead of a vector space, I'm even more confused...
Secondly, is there a way to reconcile the above algebraic definition with a geometric point of view (e.g. Lie algebra in differential geometry)? Where they say
The set of left-invariant vector fields $\mathbb g$ with the Lie bracket [ , ] : $g \times g \to g$ is called the Lie algebra of a Lie group $G$.
is there an equivalent definition in, let's say, noncommutative algebra?