Just for the sake of completeness, and in the interests of (hopefully) clarifying any confusing bits remaining, let's analyze the sample sentence.
This isn't very long, but we do still have clause embedding.
The topic, establishing the context for the rest of this sentence. We know we're talking about "that person".
Literally, "for two times / on two occasions / twice", or even "again".
The particle と has a lot of uses. In this case, it's a kind of adverbial, indicating that the 二度【にど】 portion applies to whatever comes next.
This is a set construction that basically means "to tell a lie". Literally, it's more like "to thrust out or blow out a lie".
The base verb つく has a ton of meanings. It's kind of like the English word "get" in terms of how many meanings it has. The simplest, core meaning might best translate as "to stick" -- both as a stative, when "something sticks to something else" (and is then stuck to that thing), and as an instantaneous, when "something sticks suddenly outwards", such as a stabbing or thrusting motion. In this case, this version of つく is sometimes spelled with kanji as 吐く, and it means "to expel forcefully from a narrow space". This is the same つく used for the phrase 溜息【ためいき】を吐【つ】く "to let out a sigh".
Not used so much anymore, this can be understood as basically the negative version of the Classical volitional or presumptive (that is, "want to" or "guess that") suffix む, which became the modern volitional or presumptive verb ending -(y)ō.
By way of example, しよう ("I will do, shall we do, they'll probably do") matches up with するまい ("I won't do, we shall not, they probably won't").
In this case, the と is used as a kind of quotative particle or coordinating conjunction -- this usage is similar to the use of English coordinating "that", as in, "they swore that they would do xyz..."
Past tense or perfective aspect (the action has completed) of verb 誓【ちか】う "to swear to do something".
An explanatory statement. "It is the case that [whatever came before]."
Diagramming this a bit differently to show the parsing, we might get:
The whole sentence is an explanatory statement, given the のです ending.
What we're talking about is あの人, and at it's core, what they did, which is 誓った.
What they 誓った-d about is 嘘を吐く, and と (that) they まい (wouldn't) do that 二度と (a second time / again).
Please comment if you have any questions or confusions about the above.