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Sonata-allegro form is a form with three main divisions: exposition, development and recapitulation (a coda is frequently added). It is not same as binary or ternary forms. The sonata form is a form itself.
Concerto
I'm not sure what you mean... The form of the sonata is sonata form. Or if you want to get fancy, it's First Movement Sonata-Allegro Form.
It is not certain that any one man "established" sonata-allegro form. It developed organically, over time, from the simpler a-b-a tertiary form. Embellishments on the main theme in the "b" section gradually became the more adventurous development section of the true sonata-allegro. Most of Scarlatti's keyboard sonatas did not adhere to what Haydn and Mozart would have recognized as a sonata-allegro form.
Bach
Usually there are four movements in the symphony, which is based on the sonata-cycle: Sonata-Allegro form, Allegro tempo Theme and Variations form, Andante, Adagio or Largo tempo Minuet and Trio (18th century) or Scherzo and Trio (19th century), Allegretto or Allegro tempo Sonata-Allegro, Rondo, or Theme and Variations form, Allegro, Vivace, or Presto tempo.
Schubert
the second theme
Because its crazy!
A sonata is a three-movement piece for solo piano or any solo instrument with or without piano accompaniment. Sonata-allegro refers to a particular form originally used in one-movement pieces, later incorporated in the three-movement sonata which was named for the form. A typical sonata-allegro form is I first theme, expansion II second theme, expansion III development of the opening theme or themes IV recapitulation of the themes V coda Sonata-allegro originally included tonal relationships between the first and second themes, tonal freedom during the development, and tonal unity of the recapitulation and coda.
'Allegro' means 'quite fast and lively'. There is no such thing as 'allegro form', and what I think you may mean is 'sonata allegro form', which is another version of the more common 'sonata form'. This is the form in which most first movements, and some last movements, of classical and romantic symphonies are written. The form is also used for such genres as string quartets, piano sonatas, and so on. It is described in detail elsewhere on this site.
two key areas