optics - Lensmaker's equation for thick lens in two different media

I've been looking around and I've found the lensmaker's equation for thick lens to be

$$\frac{1}{f} = \frac{n_{lens} - n_{m}}{n_m} \big( \frac{1}{R_1} - \frac{1}{R_2} + \frac{n_{lens} - n_m}{n_m} \cdot \frac{d}{n_{lens}R_1R_2}\big)$$

However I haven't been able to find an equation for a thick lens surrounded by different media such as

$$n_1 \rightarrow (n_{\rm lens}) \rightarrow n_2$$

What's the equation for the above scenario or could someone link a derivation of the thick lens equation so I could figure it out myself?