The first letters of the words of the verse,1 “I am my Beloved’s and my Beloved is mine,” form an acronym for the name Elul.2"אֲנִי לְדוֹדִי וְדוֹדִי לִי" רָאשֵׁי תֵּבוֹת אֱלוּל,
The maamar in Likkutei Torah3 beginning with this verse, explains the connection between the month of Elul and “I am my Beloved’s and my Beloved is mine”:וּמְבֹאָר בְּלִקּוּטֵי תּוֹרָה בְּהַמַּאֲמָר דִּבּוּר הַמַּתְחִיל זֶה הַּשַׁיָּכוּת דֶּאֱלוּל לַ"אֲנִי לְדוֹדִי וְדוֹדִי לִי",
that the spiritual service of Elul is that of “I am my Beloved’s,” i.e., an “arousal from below,” an initiative taken by the Jewish people to draw closer to G‑d.כִּי בֶּאֱלוּל הוּא אִתְעֲרוּתָא דִלְתַתָּא, אֲנִי לְדוֹדִי,
The beginning of the maamar in Likkutei Torah, loc. cit., explicitly states: “The implication is: The spiritual service of Elul is that of ‘I am my Beloved’s.’ ” Thus, the concept that Elul is identified with an arousal from below is an explanation of the idea that “The first letters of the words of the verse, ‘I am my Beloved’s and my Beloved is mine,’ spell out the name אלול.”
Rosh HaShanah and Yom Kippur, by contrast, reflect the second portion of the verse: “My Beloved is mine;” G‑dliness is drawn down from Above (in Kabbalistic terminology, “an arousal from Above”).וּבְרֹאשׁ הַשָּׁנָה וְיוֹם הַכִּפּוּרִים הִיא הַהַמְשָׁכָה מִלְמַעְלָה לְמַטָּה (אִתְעֲרוּתָא דִלְעֵילָּא), וְדוֹדִי לִי.
The first and second portions of the verse reflect a cause and effect relationship;
the G‑dliness drawn down on Rosh HaShanah and Yom Kippur comes as a result of “the arousal from below” on the part of the Jewish people during the month of Elul.וְהַהַמְשָׁכָה שֶׁבְּרֹאשׁ הַשָּׁנָה וְיוֹם הַכִּפּוּרִים הִיא עַל יְדֵי הָאִתְעֲרוּתָא דִלְתַתָּא שֶׁבֶּאֱלוּל.
Thus the letters of the name Elul, serve as an acronym to the entire verse: “I am my Beloved’s and my Beloved is mine,”וְזֶהוּ שֶׁאֱלוּל הוּא רָאשֵׁי תֵּבוֹת אֲנִי לְדוֹדִי וְדוֹדִי לִי
[i.e., that even the phrase “my Beloved is mine,” which refers to Rosh HaShanah and Yom Kippur, is included in the acronym for the letters of Elul].[שֶׁגַּם וְדוֹדִי לִי נִכְלָל בְּרָאשֵׁי תֵּבוֹת אֱלוּל],
For the arousal from below in Elul (“I am my Beloved’s”)כִּי הָאִתְעֲרוּתָא דִלְתַתָּא שֶׁבֶּאֱלוּל (אֲנִי לְדוֹדִי)
(also) serves as a preparation for the drawing down and revelation of G‑dliness, (“My Beloved is mine,”) on Rosh HaShanah and Yom Kippur.הִיא (גַּם) הַקְדָּמָה לְהַהַמְשָׁכָה (וְדוֹדִי לִי) שֶׁבְּרֹאשׁ הַשָּׁנָה וְיוֹם הַכִּפּוּרִים.
On this basis, the statement of the maamar — that for there to be a revelation of “My Beloved is mine” on Rosh HaShanah and Yom Kippur, it must be preceded by an arousal from below in Elul – can be understood. Although this maamar speaks about the Divine service of Elul and not Rosh HaShanah and Yom Kippur, this statement is included because it explains why “The first letters of the words of the verse, אני לדודי ודודי לי – ‘I am my Beloved’s and my Beloved is mine,’ spell out the name אלול,” i.e., the connection between “my Beloved is mine” and Elul. For the revelations of Rosh HaShanah and Yom Kippur are dependent on the arousal to teshuvah in Elul.
Explanation is, however, required: The fact that the letters vav and lamed (which allude to “my Beloved is mine”) are part of the acronym for Elul,וְצָרִיךְ בֵּאוּר, כִּי מִזֶּה שֶׁהָאוֹתִיּוֹת ו' ל' (רָאשֵׁי תֵּבוֹת וְדוֹדִי לִי) הֵם מִתֵּבַת אֱלוּל עַצְמָהּ,
indicates that Elul is not only a preparation for the drawing down of G‑dliness alluded to by the phrase “My Beloved is mine,”מַשְׁמָע, שֶׁחֹדֶשׁ אֱלוּל הוּא לֹא רַק הַקְדָּמָה לִ"וְדוֹדִי לִי",
but that the revelation of “My Beloved is mine” is also manifest within the month of Elul itself and not only afterwards, on Rosh HaShanah and Yom Kippur.אֶלָּא שֶׁבְּחֹדֶשׁ אֱלוּל הוּא גַּם הָעִנְיָן דִּ"וְדוֹדִי לִי".

Summary

The first letters of the words of the verse, אני לדודי ודודי לי — “I am my Beloved’s and my Beloved is mine,” serve as an acronym for the name אלול — Elul. The maamar in Likkutei Torah beginning with this verse explains the connection between the month of Elul and that verse: that the spiritual service of Elul is that of “I am my Beloved’s,” i.e., an “arousal from below,” an initiative taken by the Jewish people to draw closer to G‑d.
Rosh HaShanah and Yom Kippur, by contrast, are alluded to in the second portion of the verse: “My Beloved is Mine;” G‑dliness is drawn down from Above (in Kabbalistic terminology, “an arousal from Above”). Nevertheless, since the G‑dliness drawn down on Rosh HaShanah and Yom Kippur comes as a result of “the arousal from below” on the part of the Jewish people during the month of Elul, the letters of the name Elul allude to the entire verse: “I am my Beloved’s and my Beloved is mine,” i.e., they include also the revelation that takes place afterwards.
Explanation is, however, required: The fact that the letters vav and lamed (which allude to “my Beloved is mine”) are part of the word Elul, indicate that Elul is not only a preparation for the drawing down of G‑dliness alluded to by the phrase “My Beloved is mine,” but that the revelation of “My Beloved is mine” is also manifest in Elul.