Canto 2 -
Ayodhyā-kāṇḍa
Chapter 38: Rāma Requests the King to Take Care of His Mother Kausalyā
Text 2.38.11

प्रतिज्ञातं मया तावत्त्वयोक्तं देवि शृण्वता।
रामं यदभिषेकाय त्वमिहागतमब्रवीः॥

pratijñātaṁ mayā tāvat tvayoktaṁ devi śṛṇvatā
rāmaṁ yad abhiṣekāya tvam ihāgatam abravīḥ

pratijñātam = have promised; mayā = I; tāvat = only; tvayā = you; uktam = have spoken; devi = O queen; śṛṇvatā = while I listened; rāmam = to Rāma; yat = that which; abhiṣekāya = for the coronation; tvam = you; iha = here; āgatam = when He came; abravīḥ = had spoken.

“O queen, you had spoken to Rāma when He came here for the coronation. I have promised only that which you have spoken while I listened.

Kaikeyī had told Rāma this:

sapta sapta ca varṣāṇi daṇḍakāraṇyam āśritaḥ
abhiṣekam imaṁ tyaktvā jaṭājinadharo vasa

“Having given up this coronation [of Yours], with matted hair and deer skin, go to Daṇḍakāraṇya and be there for fourteen years.” (Rāmāyaṇa 2.18.37)

Daśaratha implies that he had only promised Kaikeyī the above; he had not promised Kaikeyī that Sītā-devī should also be banished.