It can be argued that with to tell, which is normally a bitransitive verb taking both sorts of objects as in to tell someone something, can exist with just an indirect object alone.
There is an unspoken “something” that is being alluded to here, and that thing unsaid is the direct object, with the person receiving the information being in the indirect object position.
However, this analysis is not universally accepted. Others view this particular situation as one where the indirect object has passed to a direct one. The OED seems to be one of these; the following citation is taken from its entry for the verb tell:
trans. to tell a person (the originally indirect or dative personal object becoming the direct).
Note the parenthetical portion. They have stopped calling it an indirect object there. It’s the only object in sight, so it “must” be a direct one. I guess.
In other languages that have stronger case systems, this sort of use can take an indirect object by itself, and when it does, it is still marked in the dative’s indirect object case. So for example, the Spanish for tell me it is dímelo; if you wanted to add and tell her too while you’re at it, you would have to say dile to mean “tell her” using the dative enclitic pronoun le. You cannot use the accusative enclitic pronoun la, because it is still indirect. So they in Spanish do not reanalyse the remaining object as a direct one. It stays dative in their minds. It is that line of reasoning that leaves it as an indirect object for some English grammarians, too.
Now, you can construct that same situation in English, but to do so you have to go back to when English was what for us today was a foreign language. A thousand years ago when English had a more robust case system, you sometimes had certain verbs that took only a dative case pronoun for their complement without an accusative case one to go along with it. So one could (perhaps) say they only took an indirect object. However, most of the time we translate those uses from Old English and dative objects into regular direct objects today so as not to confuse people.
However, one rare place where that still occurs is in the frozen relic verbs meseem and methink, once spelled me seems and me thinks (or me thought and so on). There the Old English dative me, which in those expressions came before the actual verb, became stuck when it fossilized. We no longer think of it as being an indirect object. But it really is an isolated dative, because it is the very sort of “to/for me” kind of “dative of interest” you find in modern expressions like cry me a river or sing me a song. So meseems just means it seems to me. That’s why the singular is meseems, as in Meseems unlikely that he shall pursue us meaning it seems unlikely to me.
But those are just curiosities from the museum case, not productive uses. If you abbreviate sing me a song to sing me, folks are going to think that use of me has switched to a direct object use from an indirect object one, and wonder what it is you really meant by that.
With the verb 'teach', 'calculus' will always be a direct object and 'himself' will always be an indirect object. What did he teach? Calculus. (Direct object). Whom did he teach? Himself. (Indirect object).
Other examples: "Sam hit me" or "Sam hit the table". Whom/What did she hit? Me, the table. 'whom' and 'what' have the same answer (Direct object). "Valerie told me the truth." What did she tell me? The truth (DO). Whom did she tell? Me (Indirect object). 'whom' and 'what' have a different answer. "Tom acted crazy and started talking to the TV". To whom did he start talking? To the TV (IO). "He washed his clothes" or "He washed himself". What/Whom did he wash? His clothes, himself (Direct object).
Hope this helps.
Best Answer
Promise is a very troublesome verb, syntactically speaking.
Most bitransitive communicational verbs, like tell, order, or ask, take infinitive complements with B-Equi from the Indirect Object, thus:
Sutold/ordered/askedIO[(forIO) toVP], as inThey told him to prune the peaches.
She ordered him to leave immediately.
I asked her to stop by after church.
These cases are prototype examples of B-Equi, which means that the Indirect Object of the main clause is also understood to be the Subject of the infinitive clause.
They did the telling to him, while he did the pruning. Etc.
With promise, however, the pattern breaks:
SupromisedIO[(forSu) toVP], as inShe promised me to leave right after she talked to Bill.
I promised her to take out the garbage.
Here, She did the promising, and she was to do the leaving. In other words, it's the Subject of promise that is understood to be the subject of its infinitive, and not its Indirect Object, as in the prototype cases. This is the pattern for A-Equi, where there is no Indirect Object, as in
In these A-Equi cases, She is the subject of both the main clause and the infinitive.
It feels like promise is less bitransitive than performative. Performative verbs require an audience, and promise, in particular, requires someone (even if only oneself) to attend to the promise. But that's taken for granted, just like the audiences for say, or swear, or claim, or any other performative verb. The identity of the audience of performative verbs normally need not be expressed in the sentence; this allows promise to be more comfortable with A-Equi and without an indirect object.
Promise also feels more comfortable with a that-complement, when an indirect object is present:
In a tensed clause, of course, subjects are required and aren't deleted, so no reference problem arises.