Lie-admissible algebra

Imagine you have a bunch of things that you can add together, like numbers. We call this a "set." Now, let's say we also have a special operation called "multiplication" that we can do between these things. This operation takes two things and gives us a third thing.

Now, in this set, we have something called a "Lie bracket." This is a special way of taking two things from our set and combining them to give us a third thing. This is different from our normal multiplication operation.

The Lie bracket has some special properties. First, it follows a rule called the "skew-symmetry property." This means that if we take two things, A and B, and calculate their Lie bracket, it will be the negative of the Lie bracket of B and A. In simpler terms, it's like flipping the order gives us the opposite.

Second, the Lie bracket has a property called the "Jacobi identity." This means that if we have three things, A, B, and C, and we calculate the Lie bracket of the Lie bracket of A and B with C, and add it to the Lie bracket of the Lie bracket of B and C with A, and finally add the Lie bracket of the Lie bracket of C and A with B, the result will be zero.

So, a Lie-admissible algebra is a set that has both addition and multiplication operations, and a special operation called the Lie bracket that follows these two properties. It's like having a playground with all the toys (addition and multiplication), and a special toy (Lie bracket) that has some special rules.