Let $C$ be an arbitrary algebra (more generally, a linear 1-category). The following structures are well-known:

A degree-0 product on the Hochschild cohomology $HH^*(C)$

$$
HH^*(C) \otimes HH^*(C) \to HH^*(C)
$$ ```
$$
a \otimes b \mapsto ab
$$
```

A degree-0 action of Hochschild cohomology on the Hochschild homology $HH_*(C)$

$$
HH^*(C) \otimes HH_*(C) \to HH_*(C)
$$ ```
$$
a \otimes \gamma \mapsto a\cdot \gamma
$$
```

A degree-1 unary operation on Hochschild homology (Connes differential)
$$
HH_*(C) \to HH_*(C)
$$ ```
$$
\gamma \mapsto B(\gamma)
$$
```

A degree-1 binary operation on Hochschild cohomology (Gerstenhaber bracket)
$$
HH^*(C) \otimes HH^*(C) \to HH^*(C)
$$ ```
$$
a \otimes b \mapsto a * b
$$
```

The above operations satisfy some well-known relations. (Note that I am not attempting to get the signs right.)

graded commutativity $ab = \pm ba$

more graded commutativity $a * b = \pm b * a$

Poisson identity $a * (bc) = (a * b)c + b(a * c)$

Jacobi identity $a * (b * c) + b * (c * a) + c * (a * b) = 0$

$B$ is a differential $B(B(\gamma)) = 0$

various associativities $(ab)c = a(bc)$; $(a * b) * c = a * (b * c)$; $(ab)\cdot\gamma = a\cdot(b\cdot\gamma)$

The following relation, expressing the action of a Gerstenhaber bracket on Hochschild homology in terms of the Connes differential, seems to be less well-known. At least I haven't been able to find it in the literature. $$ (a*b)\cdot\gamma = ab\cdot B(\gamma) - a\cdot B(b\cdot \gamma) - b\cdot B(a\cdot\gamma) + B(ba\cdot\gamma) $$ (Again, I haven't tried to get the signs right.)

**Question:** Is there a reference for the above relation?

Note: The above relation follows from the fact that the first homology of a certain operad space is 4-dimensional, so there must be some relation between the five degree-1 maps $HH^*(C)\otimes HH^*(C)\otimes HH_*(C)\otimes \to HH_*(C)$ which figure in the relation.

Another note: In cases where $HH^*(C) \cong HH_*(C)$ and there is a BV algebra structure, I think the relation follows from the usual definition of the Gerstenhaber bracket in terms of the BV structure. See the "Antibracket" section of this Wikipedia article.