I agree the +1 applies to a general from the same command. If it was intended to be any general, then the word "own" would not be there and the wording might be "a general" or "any general" rather than "own general" - so I believe the inclusion of the word "own" reveals the intent of the author.
Like Neil - trying a "silly idea" :
However, what if the intent of including the word "own" was to stop players claiming the bonus when an opponents general killed one of their elements? This is obviously an idiotic interpretation, but without the inclusion of the word "own" it is a possible literal interpretation of the rule. If this was the intention (to prevent your opponent unfairly claiming the +1 when your own general killed an element) then you could argue any general within your own army gives inspiration. It's not as if the rank and file wouldn't know the general of another command, and as such could be inspired by other generals from the same side inflicting damage on the enemy. But...
Assuming the previous paragraph is completely irrelevant (and the intent was not to prevent gamesmanship), but continuing to explore alternative interpretations, what is an "own" general? As Neil mentioned, is my general's general also my general? Might be a bit of a stretch.....especially given Phil's propensity for using less words, instead of more, to define a rule. For instance, to make this unambiguous the rule could say "general from own command" but it is highly probable "own general" was intended to convey the same meaning given the object of the tactical factor is the element that is in combat. Or is this circular reasoning?
In conclusion : I too think it is the general from the element's command, but am still interested to hear others opinions.
Cheers, Andrew