Learn English – Capitalization or not on fields of study in the arts

Below user Robusto's foregoing answer, user Clément requests sources which I provide here:

• An Internet search yielded this which then recommended this by the University of Sussex:

  (c) The names of languages are always written with a capital letter. Be careful about this; it's a very common mistake.

  Juliet speaks English, French, Italian and Portuguese.
  I need to work on my Spanish irregular verbs.
  Among the major languages of India are Hindi, Gujarati and Tamil.
  These days, few students study Latin and Greek.

  Note, however, that names of disciplines and school subjects are not capitalized unless they happen to be the names of languages:

  I'm doing A-levels in history, geography and English.
  Newton made important contributions to physics and mathematics.
  She is studying French literature.

• The University of Ottawa’s writing guide:

  Subject areas and disciplines

  Subject areas like biology, sociology, engineering, women’s studies and psychology are common nouns and don’t normally take a capital letter.

  The University of Ottawa has programs in an array of disciplines, from mathematics and medicine to chemistry and criminology.

  The same applies to broader areas:

  Professor Doesitall has published extensively in the humanities, pure and applied sciences and arts.

  NOTE: Capitalize the first letter in disciplines and subject areas only:

  in an official academic unit title:

  School of Psychology; Department of Theatre, etc.

  on a diploma or certificate:

  …so and so has met all of the requirements of the Master’s in Linguistics

  and in lists of official degree titles:

  The Department of Social Sciences offers these degrees: Honours Bachelor of Social Science; Master’s in Sociology…

• The University College London’s Style guide for student recruitment communications:

  • References to organisations should begin with a lower-case letter unless the title is given in full. For example: ‘The Department of Mathematics offers several degree programmes. Staff in the department specialise in...’

  • Only use initial capitalisation for job roles when singular e.g. ‘Students will be assigned a Personal Tutor. All tutors...’

  • Academic subjects begin with a lower-case letter unless part of a title. For example: ‘If you are interested in studying mathematics at UCL, a GCE A level in Mathematics is required.’

• The Boston College’s style guide:

  Departments, offices, majors

  Academic subjects are lowercase, except when you are describing a subject that is also a language, culture, or religion.

  She is a psychology major and English minor.

  He works in the psychology department and the Germanic studies department.

  The formal names of departments, offices, programs, and institutions should be capitalized; on a second or general/informal reference, lowercase.

  He works in the Department of Psychology.

  She works in the Office of Marketing Communications.

  You can pick up an application at the admissions office.

Most of the time, it is the fine arts. The only exception is when fine arts is a modifier on a noun, such as:

He has a fine arts degree.

Also, it isn't capitalized unless it is part of a title (because that follows its own rules that overrule other rules):

Jen Oak, Director, Fine Arts.