This poem contains several allusions, including a few before the poem even begins! The title, “Ariel,” makes a literary, religious, and autobiographical reference all at once.
For starters, Plath herself owned a horse named Ariel. In fact, when Plath wrote this poem, poetry-writing and horseback riding were her two morning rituals. The allusion to her own horse might help clued-in readers to figure out what's happening here; after all, the speaker never directly tells readers that this is a poem about a horseback ride. Some readers might also interpret the speaker’s ecstatic, volatile transformation as a metaphor for the process of writing poetry.
But of course, Plath probably didn't expect readers to know the name of her horse. The name "Ariel" also alludes to something that readers are more likely to be familiar with: Shakespeare's great late play, The Tempest. In this play, the magician Prospero finds an air-spirit named Ariel imprisoned in a tree by an evil witch. Prospero releases Ariel on the condition that Ariel serve him for a year before going free (though as it happens, Prospero is reluctant to keep his promise at first, wanting to cling to Ariel's magic powers for his own purposes).
The poem's allusion to the play thus suggests both the liberated freedom of the air-spirit and the speaker's "imprisonment" in daily life and the material world at the beginning of the poem.
Finally, "Ariel" is a Hebrew name for Jerusalem—a name that literally translates to "Lion of God.” The speaker reinforces this allusion when she refers to the horse as “God’s lioness.” This allusion plays up the horse’s might and spiritual power, while subtly linking these qualities with femininity.
The poem’s final allusion appears when the speaker refers to herself as “White / Godiva. As the legend has it, Lady Godiva, a medieval noblewoman, repeatedly begged her husband to lift the oppressive taxes that he had imposed on his struggling people. He finally agreed to do so, but only if she rode through town on horseback—naked. After ordering everyone to stay inside and close their windows, Lady Godiva took her husband up on his offer.
This allusion resonates with the strength, power, and potential danger that the speaker feels when she sheds her physical limitations: much like Lady Godiva, she removes her "outer layers" to stand up for what she truly feels.