Einstein Summation Convention: One as Upper, One as Lower?

In the 'strict' sense, you should only apply the summation convention to a pair of indices if one is raised and another is lowered.

For example, consider a vector $v$ and a dual vector $f$ (i.e. a map from vectors to numbers). Then one can compute $f(v)$, the number that results from $f$ acting on $v$. In components, this would be written as $f_i v^i$, since dual vectors have lower indices.

If, instead, you have two vectors $v$ and $w$, there is generally no way to combine them into a number, and the quantity $v^i w^i$ makes no sense. But if you have a metric $g_{ij}$, you can use it to turn $w$ from a vector into a dual vector, with new components $g_{ij} w^j$. Then you can act with this dual vector on $v$, giving $g_{ij} v^i w^j$. Note that all indices are paired correctly.

That being said, there are lots of exceptions:

• A lot of field theory texts and even GR texts will write $v^i w^i$, but you're supposed to remember it really means $g_{ij} v^i w^j$. When you do explicit computations, you have to put that factor in yourself.
• If you're working in a space with a simple metric (like Euclidean space, where $g_{ij}$ is the identity), texts might omit $g_{ij}$ because it doesn't "do anything". That is, the vector $w^j$ and corresponding dual vector $g_{ij} w^j$ always have the exact same components, so they might as well identify them.
• If the previous point is true, the author might choose to use index position to store some other kind of information, so the summation convention remains 'strict'. This happens more often in non-physics texts.

There are enough possible conventions that you should just check the front of the book every time.

There is nothing wrong by summing up indices when both indices are either up or down. It is just a matter of convention. However the meanings can be different if you are in a Relativistic theory.

When you sum one up and one down indices in Relativity it means you have a Lorentz invariant quantity because you are combining covariant and contravariant components in such a way that the combination does not change under Lorentz Transformations. This is a (nice) rule adopted by most authors and its importance is in the fact we can immediately identify invariant quantities. A few authors though use always down indices even for relativistic theories. For instance, Rubakov's Classical Theory of Fields.

If you are in Euclidean space, there is nothing to worry about. Normally people use all indices down.