Examples of unital non-associative algebras

Question

What is an example of a unital non-associative algebra?

Answer 1

A pre-Lie algebra structure on $\mathfrak{gl}_2(\Bbb C)$ is given as follows, for an arbitrary $a\in \Bbb C$, \begin{align*} y_1\cdot y_2 & = \frac{a+1}{2}y_3+\frac{1}{2}y_4, & y_2\cdot y_4 & =(1-a)y_2, & y_4\cdot y_1 & =(a+1)y_1,\\ y_1\cdot y_3 & = -y_1, & y_3\cdot y_1 & =y_1, & y_4\cdot y_2 & =(1-a)y_2,\\ y_1\cdot y_4 & = (a+1)y_1, & y_3\cdot y_2 & =-y_2, & y_4\cdot y_3 & =(1-a^2)y_3-ay_4,\\ y_2\cdot y_1 & = \frac{a-1}{2}y_3+\frac{1}{2}y_4, & y_3\cdot y_3 & =ay_3+y_4, & y_4\cdot y_4 & =a(a^2-1)y_3+(a^2+1)y_4,\\ y_2\cdot y_3 & = y_2, & y_3\cdot y_4 & =(1-a^2)y_3-ay_4, & & \end{align*} This $4$-dimensional algebra $(A,\cdot)$ is not associative. Now adjoin a unity element to $A$ to obtain a nonassociative, unital algebra - see here, page $8$ after $(7')$.

Answer 2

Some Jordan algebras are a good example:

$$a\circ b = \frac{a b + b a}{2}$$ starting from an associative algebra $(A, \cdot)$ with unit $1$.

Note: Jordan was interested in an operation on hermitian operators. So start with $(A, \circ)$ where say $A = M_n(\mathbb{C})$, and then $\mathcal{H}$ the hermitian matrices are a Jordan subalgebra.