How do you determine whether the adverbial form of an adjective should end in -ly or -ally?
For most adjectives, the reason to use -ally, rather than -ly, is that you’re adding -ly to a noun plus the inflection -al, which forms the adjective, as in emotionally, musically, or traditionally. You’re starting with emotion, music, or tradition, converting the noun to an adjective (emotional, musical, or traditional), and then adding -ly.
But this process isn’t consistent. The adjectival form of romance is romantic, not romantical, nor are academical or sarcastical adjectival forms. But you can distinguish these exceptions by noting that the noun form of these words is not the word minus -al; those forms are the adjectival ones, and the noun forms are irregular: Romantic is derived from romance, not romant; academic comes from academy, not academ; and sarcastic stems from sarcasm, not sarcast.
A generalization is that -ally follows words that end with the letter c; however, the adverbial form of public is publicly, not publically, and there may be other exceptions. In addition, some words bereft of the letter c, like sentimentally, are anomalously constructed.
Note, too, that other major parts of speech include words that end in -ly: for example, the nouns ally and bully, the adjectives friendly and lonely, and the verbs apply and supply. (Also, adjectives ending in -ly have no adverbial form.)
To summarize, if an adjective ends in -al, append -ly to produce its adverbial form. If an adjective does not end in -al, attach -ly — without inserting -al first — to transform it into an adverb.