Direct SAT to 3-SAT reduction
Each SAT clause has 1, 2, 3 or more variables. The 3 variable clause can be copied with no issue

The 1 and 2 variable clauses {a1} and {a1,a2} can be expanded to {a1,a1,a1} and {a1,a2,a1} respectively.

The clause with more than 3 variables  {a1,a2,a3,a4,a5} can be expanded to {a1,a2,s1}{!s1,a3,s2}{!s2,a4,a5} with s1 and s2 new variables whose value will depend on which variable in the original clause is true