I think elements can be broken down, but their definition says otherwise, why?
What definition, where? The short answer is given in the IUPAC Gold Book:


  chemical element
  
  
  A species of atoms; all atoms with the same number of protons in the atomic nucleus.
  A pure chemical substance composed of atoms with the same number of protons in the atomic nucleus. Sometimes this concept is called the elementary substance as distinct from the chemical element as defined under 1, but mostly the term chemical element is used for both concepts.
  


A longer answer is: Yes, physicists have managed to split atoms further down than to nuclei and electrons. It is also possible to "transmute" an element into another, such as creating Plutonium from Uranium. This involves a lot of machinery and energy, whereas splitting a substance into its constituent elements can be as simple as heating or dissolving in an acid and could be done in an afternoon with a few glass beakers. 

Looking at it from a more theoretical perspective, one can (snarkily) define Chemistry as "the Physics concerned with the outer electron shell". Once you work on other areas of the atom (that is, the nucleus) or an electron shell of an atom can no longer be reasonably defined (plasma as in the sun), you tend to leave the area of chemistry (and the applicability of the term "element").