Adverb clauses of purpose are introduced by the subordinating conjunctions that, so that, in order that and lest.
- We eat that we may live.
- He works hard so that he will become a millionaire.
- Put on your warm clothes lest you should catch a chill.
- Schools were closed early in order that students might reach home before the thunderstorm.
In an informal style, so that is more common than in order that.
These expressions are usually followed by modal auxiliary verbs such as will, can or may.
- She wants to study in England so that she can perfect her English.
- We are starting now so that we will reach there before sunset.
In an informal style that can be dropped after so; this is very common in American English.
- I have come early so that I can meet you. OR I have come early so I can meet you.
Lest means that…not, and, therefore, it will be wrong to add another not in the following clause. Moreover it should be noted that the only auxiliary verb that can be used after lest is should.
- Reserve your tickets early lest you miss the chance. OR Reserve your tickets early lest you should miss the chance. (NOT Reserve your tickets early lest you do not miss the chance.)