Why Were Nadav and Avihu Killed?

Introduction

An Undefined Sin

Parashat Shemini opens with a description of the eighth day of the Tabernacle's consecration.  As the service is completed and Hashem's glory descends, the climactic moment is marred by the tragic deaths of Nadav and Avihu:

EN/HEע/E
(א) וַיִּקְחוּ בְנֵי אַהֲרֹן נָדָב וַאֲבִיהוּא אִישׁ מַחְתָּתוֹ וַיִּתְּנוּ בָהֵן אֵשׁ וַיָּשִׂימוּ עָלֶיהָ קְטֹרֶת וַיַּקְרִיבוּ לִפְנֵי י"י אֵשׁ זָרָה אֲשֶׁר לֹא צִוָּה אֹתָם. (ב) וַתֵּצֵא אֵשׁ מִלִּפְנֵי י"י וַתֹּאכַל אוֹתָם וַיָּמֻתוּ לִפְנֵי י"י.
(1) And Nadab and Abihu, the sons of Aaron, took each of them his censer, and put fire therein, and laid incense thereon, and offered strange fire before the Lord, which He had not commanded them. (2) And there came forth fire from before the Lord, and devoured them, and they died before the Lord.

The verses attribute the brothers' deaths to bringing "foreign fire", but do not explain to what this refers or why it was problematic. If Nadav and Avihu simply brought incense, a commanded daily offering, why is it called "foreign"?  Moreover, why would such an action be punished as a capital crime, especially in the midst of celebrating the inauguration of the Mishkan?

By My Close Ones I Will be Sanctified

After the tragedy, Moshe tells Aharon: "הוּא אֲשֶׁר דִּבֶּר י"י לֵאמֹר בִּקְרֹבַי אֶקָּדֵשׁ" ("This is what Hashem said: by those close to me, I will be sanctified").  The statement is ambiguous. To whom does the phrase "my close ones" refer?  How was Hashem to be sanctified through them?  If Nadav and Avihu are the subject of these words, it suggests that even in death they were considered holy.  If so, though, why were they punished so severely?  Should they be considered sinful or beloved?

Additionally, when was it that Hashem said this?  Nowhere prior to this incident is there explicit evidence of any such comment.  Is it just to be assumed or are Hashem's words hinted to elsewhere?1  If the latter, how might the original context of Hashem's remarks shed light on Nadav and Avihu's actions?

Additional Questions

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