Learn Icelandic: Ég sé vanræksluna, þótt byggingin sé falleg. - I see the neglect, although the building is beautiful.

The prefix van- indicates a lack of something, doing something poorly, or "under-doing" it. Here, it attaches to ræksla, a noun related to the verb rækja (to tend to or cultivate). So, vanræksla literally means "under-tending" or "neglect".

Why does the word sé appear twice in this sentence, and do they mean the same thing?

They are actually two completely different verbs! The first sé is the first-person present indicative of sjá (I see). The second sé is the third-person present subjunctive form of vera (to be), translating to "is".

Why do we use the subjunctive sé for the building instead of the normal indicative er?

The conjunction þótt (although) almost always triggers the subjunctive mood in Icelandic, regardless of whether the statement is a proven fact. So even though the building is definitely beautiful, the grammar requires sé instead of er after þótt.

What case is vanræksluna, and why does it have the definite article?

It is in the accusative case because it is the direct object of the verb sjá. It has the feminine definite article -na attached (vanræksla + -na). While English might leave abstract concepts like "neglect" indefinite, Icelandic often uses the definite article when referring to a specific, tangible instance of it, like the specific neglect of this building.