Realizing the FLEECE vowel as [ij] or [ɪi̯]

All tense vowel phonemes in English have inconsequential phonetic offglides at their ends. This is just as true of /i/ and /u/ as it is of /e/ and /o/.

It doesn't change which vowel is perceived to have been said, so we seldom bother writing it out except for in the very narrowest of [phonetic] transcriptions and nearly always ignore these in broad /phonemic/ ones.

Because these are little phonological effects that are applied automatically and almost impossible to suppress, we don't even think about them, let alone write them down. They happen because of how we articulate English: most of us never maintain the same level tenseness of the vowel throughout its entire articulation. We relax or tighten it a bit on the way out, or sometimes on the way in. Or both. It isn’t usually constant.

(This strongly contrasts with languages like Spanish and Italian, where you can have a tense vowel that stays tense always. Spanish has only tense/close vowels, although Italian also has a pair of lax/open ones.)

The English Wikipedia article on diphthongs represents these in broad terms in this fashion:

Standard English diphthongs

Word	English diaphoneme	RP (British)	Australian	GenAm	Canadian
low	//oʊ//	[əʊ̯]	[əʉ̯]	[o̞ʊ̯]¹	[o̞ʊ̯]¹
loud	//aʊ//	[aʊ̯]	[æɔ̯]	[aʊ̯~æʊ̯]	[aʊ̯~æʊ̯]²
lout	//aʊ//	[aʊ̯]	[æɔ̯]	[aʊ̯~æʊ̯]	[ʌʊ̯]³
lied	//aɪ//	[aɪ̯]	[ɑɪ̯]	[äɪ̯]⁴	[äɪ̯]⁴
light	//aɪ//	[aɪ̯]	[ɑɪ̯]	[ʌɪ̯]³	[ʌɪ̯]³
lay	//eɪ//	[eɪ̯]	[æɪ̯]	[eɪ̯]¹	[eɪ̯]¹
loin	//ɔɪ//	[ɔɪ̯]	[oɪ̯]	[ɔɪ̯]	[ɔɪ̯]
loon	/uː/⁵	[ʊu̯]	[ʉː]	[ʉu̯]	[ʉu̯]
lean	/iː/⁵	[ɪi̯]	[ɪi̯]	[i]	[i]
leer	//ɪər//	[ɪə̯]	[ɪə̯]⁶	[ɪɹ]	[ɪɹ]
lair	//ɛər//	[ɛə̯]⁷	[eː]	[ɛɹ]	[ɛɹ]
lure	//ʊər//	[ʊə̯]⁷	[ʊə̯]	[ʊɹ]	[ʊɹ]

1. In Scottish, Upper Midwestern, and California English, /oʊ̯/ is monophthongal [oː].
2. In Pittsburgh English, /aʊ̯/ is monophthongal [aː], leading to the stereotypical spelling “Dahntahn” for “downtown”.
3. Canadian English and some dialects of northern American English exhibit allophony of /aʊ̯/ and /aɪ̯/ called Canadian raising – in some places they have become separate phonemes. GA and RP have raising to a lesser extent in /aɪ̯/.
4. In several American dialects such as Southern American English, /aɪ̯/ becomes monophthongal [aː] except before voiceless consonants.
5. The erstwhile monophthongs /iː/ and /uː/ are diphthongized in many dialects. In many cases they might be better transcribed as [uu̯] and [ii̯], where the non-syllabic element is understood to be closer than the syllabic element. They are sometimes transcribed /uw/ and /ij/.
6. Most Australian English speakers monophthongize “-ee-” vowels. However, Western Australian English is an exception, as it generally features centring diphthongs in words like fear and beard. See: Macquarie University, 2010, Regional Accents (30 January 2015).
7. In Received Pronunciation, the vowels in lair and lure may be monophthongized to [ɛː] and [oː] respectively (Roach (2004:240)).

John Lawler pointed out the same thing which we see Wikipedia mention in their footnote 5, where they explain that /i/ and /u/ might be better transcribed as [uu̯] and [ii̯] and which are sometimes transcribed /uw/ and /ij/, when he wrote in a comment:

Oh, it's quite standard and has even been used for Trager-Smith phonemic notation. If you use /iy/ for [i] and /i/ for [ɪ], you can use just ASCII for tense/lax vowels.

Unfortunately, that shows only one version for each broad set that itself comprises numerous dialects within its umbrella. In practice, they’re all a little different in each of those dialects.

For details like that you can consult the online Sound Comparisons website, which includes both narrow phonetic transcriptions as well as actual recordings of native speakers speaking each word in isolation. I recommend you consult their numerous renditions for such words as:

• see and three
• two and new
• toe and snow
• day and naked
• all and wash

Because those narrow transcriptions are nearly impossible for non-specialists to decipher, here are some examples they provide along with what each symbol means:

see [sɪiˑ]

 s      voiceless alveolar sibilant             U+0073  LATIN SMALL LETTER S
 ɪ      near-close near-front unrounded vowel   U+026A  LATIN LETTER SMALL CAPITAL I
 iˑ     close front unrounded vowel             U+0069  LATIN SMALL LETTER I
        half-long                               U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON

three [θɹɪ̠ˑiˑ]

 θ      voiceless dental fricative              U+03B8  GREEK SMALL LETTER THETA
 ɹ      voiced alveolar approximant             U+0279  LATIN SMALL LETTER TURNED R
 ɪ̠ˑ     near-close near-front unrounded vowel   U+026A  LATIN LETTER SMALL CAPITAL I
        retracted                               U+0320  COMBINING MINUS SIGN BELOW
        half-long                               U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON
 iˑ     close front unrounded vowel             U+0069  LATIN SMALL LETTER I
        half-long                               U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON

two [tʰʊ̟uˑ]

 tʰ     voiceless alveolar plosive              U+0074  LATIN SMALL LETTER T
        aspirated                               U+02B0  MODIFIER LETTER SMALL H
 ʊ̟      near-close near-back rounded vowel      U+028A  LATIN SMALL LETTER UPSILON
        advanced                                U+031F  COMBINING PLUS SIGN BELOW
 uˑ     close back rounded vowel                U+0075  LATIN SMALL LETTER U
        half-long                               U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON

two [tʰʉ̞ÿˑ]

 tʰ     voiceless alveolar plosive              U+0074  LATIN SMALL LETTER T
        aspirated                               U+02B0  MODIFIER LETTER SMALL H
 ʉ̞      close central rounded vowel             U+0289  LATIN SMALL LETTER U BAR
        lowered                                 U+031E  COMBINING DOWN TACK BELOW
 ÿˑ     close front rounded vowel               U+0079  LATIN SMALL LETTER Y
        centralized                             U+0308  COMBINING DIAERESIS
        half-long                               U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON

new [njʊuˑ]

 n      voiced alveolar nasal                   U+006E  LATIN SMALL LETTER N
 j      voiced palatal approximant              U+006A  LATIN SMALL LETTER J
 ʊ      near-close near-back rounded vowel      U+028A  LATIN SMALL LETTER UPSILON
 uˑ     close back rounded vowel                U+0075  LATIN SMALL LETTER U
        half-long                               U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON

new [nɵ̜ˑÿˑ]

 n      voiced alveolar nasal                   U+006E  LATIN SMALL LETTER N
 ɵ̜ˑ     close-mid central rounded vowel         U+0275  LATIN SMALL LETTER BARRED O
        less rounded                            U+031C  COMBINING LEFT HALF RING BELOW
        half-long                               U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON
 ÿˑ     close front rounded vowel               U+0079  LATIN SMALL LETTER Y
        centralized                             U+0308  COMBINING DIAERESIS
        half-long                               U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON

toe [tʰʌ̝̈oˑ]

 tʰ     voiceless alveolar plosive              U+0074  LATIN SMALL LETTER T
        aspirated                               U+02B0  MODIFIER LETTER SMALL H
 ʌ̝̈      near-open central unrounded vowel       U+028C  LATIN SMALL LETTER TURNED V
        raised                                  U+031D  COMBINING UP TACK BELOW
        centralized                             U+0308  COMBINING DIAERESIS
 oˑ     close-mid back rounded vowel            U+006F  LATIN SMALL LETTER O
        half-long                               U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON

toe [tʰœ̈ˑÿ]

 tʰ     voiceless alveolar plosive              U+0074  LATIN SMALL LETTER T
        aspirated                               U+02B0  MODIFIER LETTER SMALL H
 œ̈ˑ     open-mid front rounded vowel            U+0153  LATIN SMALL LIGATURE OE
        centralized                             U+0308  COMBINING DIAERESIS
        half-long                               U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON
 ÿ      close front rounded vowel               U+0079  LATIN SMALL LETTER Y
        centralized                             U+0308  COMBINING DIAERESIS

day [dɛ̝ˑɪ]

 d      voiced alveolar plosive                 U+0064  LATIN SMALL LETTER D
 ɛ̝ˑ     open-mid front unrounded vowel          U+025B  LATIN SMALL LETTER OPEN E
        raised                                  U+031D  COMBINING UP TACK BELOW
        half-long                               U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON
 ɪ      near-close near-front unrounded vowel   U+026A  LATIN LETTER SMALL CAPITAL I

naked [ˈnɛ̞ˑɪkɪd]

 ˈ      primary stress                          U+02C8  MODIFIER LETTER VERTICAL LINE
 n      voiced alveolar nasal                   U+006E  LATIN SMALL LETTER N
 ɛ̞ˑ     open-mid front unrounded vowel          U+025B  LATIN SMALL LETTER OPEN E
        lowered                                 U+031E  COMBINING DOWN TACK BELOW
        half-long                               U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON
 ɪ      near-close near-front unrounded vowel   U+026A  LATIN LETTER SMALL CAPITAL I
 k      voiceless velar plosive                 U+006B  LATIN SMALL LETTER K
 ɪ      near-close near-front unrounded vowel   U+026A  LATIN LETTER SMALL CAPITAL I
 d      voiced alveolar plosive                 U+0064  LATIN SMALL LETTER D

wash [wɔ̞ˑʃ]

 w                                              U+0077  LATIN SMALL LETTER W
 ɔ̞ˑ     open-mid back rounded vowel             U+0254  LATIN SMALL LETTER OPEN O
        lowered                                 U+031E  COMBINING DOWN TACK BELOW
        half-long                               U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON
 ʃ      voiceless postalveolar fricative        U+0283  LATIN SMALL LETTER ESH

all [ɔˑəɫ]

 ɔˑ     open-mid back rounded vowel             U+0254  LATIN SMALL LETTER OPEN O
        half-long                               U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON
 ə      mid-central vowel                       U+0259  LATIN SMALL LETTER SCHWA
 ɫ      velarized alveolar lateral approximant  U+026B  LATIN SMALL LETTER L WITH MIDDLE TILDE