I don't think its related to the show in anyway, I find it more related to Azazel and ahrimanI'd like to say it's silly, but It does have merit. My first thought that comes to mind in regards to a "Black Flame" is Amaterasu from Naruto. Although you may not be familiar, in the show it is depicted as a unending Black Flame as hot as the sun. This ties back into the Japanese deity of the same name.
Now I myself have no knowledge of Eastern Religion, Occult Teachings, or practices. But, that might be a good look for you.
black flame is confusing there's an Azazel black flame Lucifer's black flame ahrimans black flame, there many more spirits related to it I could stanias Azazels chant of black flame in youtube to be most well known and accessibleTo me, the Black Flame is roughly the Luciferian analogue to Crowley’s True Will. There are important differences, but at 10k feet, it is a useful shorthand comparison for an explanation.
It is the individual divine spark: the truest core of the self. To fan the Black Flame is to learn, forge, refine, and become more fully what you are. In that sense, it is part of the process of self deification.
For me, the Black Flame is my fire, my stance, the shape of my fire before voice is added to it. It is the core essence from which my choices and actions emerge.
The work is to find it, strip away what is not truly yours, forge what remains, and live in alignment with it. In other words, it is not merely a technique or temporary state. It is the path, and ultimately the life.
Your explanation might be different. This isn't an official explanation of Luciferian theology, just a quick reply. Im pretty tired so this is what ya get for now.
Finally someone giving straight forward answer, thank youYou can treat the black flame as your magical power, especially for left-hand path practitioners. The right-hand path equivalent would be the light of creation. the stronger your black flame is, the stronger the impact of your spells would be. On a more detailed note, it is a power of transgression. People like Anton Lavey made vague notions about it, such as how it represents enlightenment, the drive within human beings etc. But that is looking at black flame from a psychological perspective rather than as a proper occult phenomenon. From my research and my own experiences, it is the power that allows us to enforce our own will even if it goes against the laws of the ordered universe. Hence, it is a transgressive, adversarial power. For example, trying to conjure flames out of thin air would be considered a transgression of the law of nature. It is a microcosmic reflection of the black sun (probably the very black sun associated with the Nazis and which is supposedly in the Qlipha of Thagirion). This power can help you bend nature and its laws, and other spirits to your will the more it strengthens. Left-hand path magicians typically wield such power because of their mindset where they want to achieve self-deification (unless they are anti-cosmic) and be able to impose their own will upon the world rather than remain subservient to another power.