You can use the Glyph of Warding (p. 245 PHB) to store Planar Binding (p. 265 PHB) and/or Magic Circle (p. 256 PHB) in order to bind "a celestial, an elemental. a fey. or a fiend to your service." Cast either Infernal Calling or Summon Greater Demon on top of the glyph to trigger it.
A spell glyph can be inscribed on any surface. It can store any spell of the Glyph's spell level or lower which targets a single creature (Planar Binding) or an area (Magic Circle), or which summons hostile creatures or traps (specifically used as an example in the spell). If the target of the Planar Binding ward fails their save, they are bound. When the Glyph is triggered, the stored spell is cast. Concentration spells last their full duration regardless, as explicitly stated under the "Spell Glyph" entry.
There is little question either RAW or RAI whether a Glyph can be used in this way. This would allow a single mage to planar bind summoned creatures given time and resources.
To do this:
- Cast Glyph of Warding - Spell Glyph to store a casting of Planar Binding.
- Cast Glyph of Warding - Spell Glyph to store a casting of Magic Circle for safety.
Cast Summon Greater Demon/Infernal Calling either right on top of the glyphs or outside the glyphs (later commanding them to give their true name and walk onto the glyphs)
Optional: Cast Feeblemind on the demon/devil to tank its Charisma saving throw to guarantee success.
Also, Planar Binding reads "The creature must be within range for the entire casting of the spell." This could be interpreted to mean that you cannot planar bind via a Glyph, but there is a way around this problem.
- Cast Glyph of Warding - Spell Glyph to store Summon Greater Demon (Summons are expressly allowed to be stored within the Glyph's text) and set the spell to trigger on a command word.
- Cast Glyph of Warding - Spell Glyph to store Magic Circle for safety and set it to trigger once a demon enters the area.
- Trigger the first glyph with your verbal command. It doesn't matter if you don't control the demon - the Magic Circle will contain it regardless. Immediately cast Planar Binding to bind the demon.
If timing is STILL deemed to be an issue for your DM (the summon lasts 1 hour and Planar Binding has a 1 hour cast time), 2 levels of Sorcerer can snag you the Extended Spell metamagic to prolong the summoning to allow you to bind the demon.
This trick also works with Gate. Use Gate to summon a powerful celestial, elemental, fey, or fiend (preferably something without legendary resistances - something like a Red Abishai or a Goristro, for example). Magic Circle can prevent them from leaving the plane unless they pass a Cha check. Slap them with Feeblemind to tank their saving throws and hard-cast Planar Binding.
ALTERNATE 2: Use a Simulacrum at level 13 to assist you.
Case 1) The spell ends and the elemental disappears:
The clause about concentration does not apply, the elemental exists as the effect of an active spell, if the magic is dispelled the elemental goes.
Case 2) There is no spell to dispel, the elemental is unaffected:
The spell has already ended as concentration has been broken and Conjure Elemental is a concentration spell.
Conjure Elemental (PHB p.225)
Duration: Concentration, up to 1 hour
Concentration (PHB p.203):
Some spells require you to maintain concentration in order to keep their magic active. If you lose concentration, such a spell ends.
Therefore while the elemental persists for an hour after the spell ends, it is no longer part of an active spell effect so there is nothing to dispel, the elemental became free of the summoning spell when concentration was lost.
Another way of looking at it is after the spell ends due to loss of concentration, the elemental exists as an environmental change rather than a spell effect. It can't be dispelled any more than the environmental effects of a delayed blast fireball (PHB p.231) can be dispelled following loss of concentration.
Jeremy Crawford had this to say about concentration with respect to conjure elemental:
When you willingly stop concentrating on a concentration spell, the spell ends.
Conjure Elemental: "The elemental disappears when it drops to 0 hit points or when the spell ends."
This is him saying that if you deliberately end your concentration the spell ends and the Elemental disappears.
He also had this to say about concentration in general:
Willingly ending your concentration on a spell in D&D isn't the same thing as having your concentration broken.
This spell states:
If your concentration is broken, the elemental doesn't disappear.
If you have your concentration broken the spell ends, as per the concentration rules which are not overridden by any specific rule in the spell, but the effect of the spell ending is overridden by the fact that the elemental does not disappear along with the spell in this specific case.
I also believe the RAI of this spell is that if you have your concentration broken the elemental becomes a free agent not bound by the summoning magic in any way, and thus it is always dangerous undertaking, where you can't even dispel your own magic.
Best Answer
RAW, no matter our answers, the result is (nearly) the same!
Your question is an excellent one, and a fascinating edge-case interaction between two spells. But as surprising as it may seem, no matter our answer the outcome will be the same: whether or not the original Conjure Elemental spell is "extended" (or, "refreshed"), the elemental in question will remain on this plane for the 24 hour duration of Planar Binding, under the control of Planar Binding's caster! The one difference is that in one case, the Elemental will disappear after the 24 hour duration, and in another it may not. Let's consider the details below:
For the purposes of this question, let's assume that the caster of Conjure Elemental (CE) may not be the same person who cast Planar Binding (PB): designate a Warlock as the one who cast CE (via an Invocation), and a Wizard as the one who cast PB. Let us also assume that the Conjure Elemental spell was cast 40 minutes ago, then the caster of that spell lost concentration and Planar Binding was immediately cast by the Wizard casting Wish, precisely emulating the Planar Binding spell (so that we don't run into problems with PB's usual 1 hour casting time, which normally would mean the elemental disappeared before PB was done being cast). Now we can consider the two possible cases.
Case 1.) The Conjure Elemental spell's duration is "extended", and thus is still in effect for 24 hours
It's debatable whether this makes sense. Can you "extend" a spell that is already over? For the sake of argument, we'll ignore that question for the moment (returning to it later), and focus for now on the mechanical effect of this "extension."
First of all, note that the Warlock is not forced to concentrate on the Conjure Elemental spell for the next 24 hours. Concentration is part of a spells duration. Planar Binding has changed the duration of Conjure Elemental " to match the duration of" Planar Binding (PHB, p. 265, Planar Binding). Thus, Planar Binding did not extend the Conjure Elemental spell to "Concentration, up to 24 hours": it extended it to "24 hours". So the extended Conjure Elemental spell no longer requires concentration on the part of the Warlock (or the Wizard).
Normally, the Warlock who cast CE would be in control of the Elemental for the duration of CE. But "specific overrides general," so while PB is in effect, the Wizard will control the elemental. And as per the text of Conjure Elemental (PHB, p. 225, bold added):
Thus, at the end of the 24 hours (from when PB was cast), the CE spell will end, and the elemental will disappear.
Thus, if the Conjure Elemental spell is "extended" by Planar Binding:
Now let's consider the other case.
Case 2.) The Conjure Elemental spell can't be "extended," and Planar Binding is the only currently active spell
Now in this case, the elemental will still obey the Wizard who cast PB, since according to the text on Planar Binding (PHB, p. 265):
Note that if the Wizard is an ally of the Warlock who cast Conjure Elemental (and lost control of it), the elemental may be hostile to the Wizard. But that's a minor point.
The important question at this point is how long does the Elemental actually remain on the Wizard's plane? It will remain in your service for 24 hours, the Elemental could still be bound to your service when a magical effect whisks it away to another plane. As strange as it may seem for a spell called "Planar Binding" (PB), the PB spell doesn't specify that it keeps a creature on any particular plane. A fire elemental could be shunted back to the elemental plane of fire still under your control. It might just stand there for PB's duration going "well, master will surely give me instructions soon."
So the question becomes, will the elemental remain longer than the next 20 minutes? Let's look at the text of Conjure Elemental for guidance. First, we know that it will normally disappear "when the spell ends." But as you noted, there is an exception (PHB, p. 225):
Since the Warlock lost concentration on CE, the elemental doesn't disappear (although the spell has ended). Then, the Wizard took control of the elemental via Planar Binding. You might think that the elemental will disappear in twenty minutes (since it's been 40 minutes since it was summoned originally). But the clause in the final sentence only applies to uncontrolled elementals: and the elemental is no longer uncontrolled! It is currently controlled by the Wizard. And the original clause that the warlock's "concentration was broken" still holds, so it "doesn't disappear"!
Thus, in this second case where the Conjure Elemental wasn't "extended":
Same as before! Except... then what happens?
So if Conjure Elemental wasn't extended, what happens when Planar Binding wears off?
Now we come to the one tricky part. Then 24 hours later, the Wizards control of the elemental will end. Now, the elemental is once again an "uncontrolled elemental", and is subject to the clause in the previous Conjure Elemental spell. As an uncontrolled elemental, it disappears "one hour after ...[it was] summoned." Strictly speaking, 24 hours (and forty minutes) is not the same thing as "one hour after." Some DMs might rule that any time more than an hour after something counts as "one hour after" it. Others might rule that there is a difference between "one hour after" and "after one hour" (the later of which would definitely include all times greater than an hour afterwards).
After the 24 hours are up, the Elemental may simply remain on this plane, hostile to either the Warlock, the Wizard, or both. Or it may disappear if the DM decides that "one hour after" includes the time later than an hour after, or decrees that the elemental's disappearance was suppressed while it was controlled, but is now activated again. The safest option for the caster of Planar Binding would be to give the elemental an order to destroy itself as quickly as possible (without harming anyone else) in the last few minutes of that spell's duration (for example, ordering a Fire Elemental to jump off a cliff into the ocean and swim as far away as possible). This should make the one point of uncertainty mostly moot.
So which happens? Case 1 or Case 2?
Unfortunately, your DM will have to decide. One could argue that the elemental definitely "was brought by another spell" so that spell's duration is now extended. But on the other hand, you could argue that even if a spell's duration is extended, that doesn't change whether or not the spell is currently over. But you could retort that it does, since the Conjure Elemental spell is no longer concentration, which is what caused it to end in the first place. But then you could say that a spell that is no longer active can't be "extended," because there is nothing to extend. And so on back and forth.
Personally, I suspect that Case 1 (where CE is "extended") is more likely to be in line with the intended function of Planar Binding, and also more likely to create a simple situation with no ambiguities, so as a DM it's the case I'd be more likely to chose. But ultimately, it will be up to your DM to decide which of these two cases will apply. But in either case, the results will be surprisingly similar.