This is a follow-up to my question [here][1].

 - For **regular subalgebras** of some group's Lie algebra the root system of the subalgebra is a subset of the root system of the original's group algebra. In other words, the generators of the subalgebra are a subset of the generators of the original group.
 - Subalgebras whose root system is not a subset of the root system of the original algebra are called **special subalgebras**. Therefore, the generators are not a subset of the original's group generators.

Using the Higgs mechanism, as explained nicely by @Heterotic in the question I linked to above, we simply check which generators remain unbroken after one or more Higgs fields get a vev. Then:

> "The remaining subgroup after the symmetry breaking is simply the
> group generated by the unbroken generators."

This is certainly correct for regular subalgebras, but how does this work for special subalgebras? How can we determine that we broke to a special subalgebra by giving a vev to a Higgs field, if the generators of the subalgebra are not a subset of the original's group generators?

  [1]: http://physics.stackexchange.com/questions/192274/how-to-find-the-remaining-subgroup-after-some-higgs-field-gets-a-vev#comment408980_192274