# Writing out equation for linear mixed model

I have a linear mixed model with two random effects. In R it looks like this:

lmer(y ~ x1 + x2 + x3 + x2:x3 + (1|Plot) + (1|ID))


I figure for a fixed-effect-model the equation would be:

But how do I account for the two random effects?

$$y_{ij} = \beta_{0} + \beta_{1}X_{1ij} + \beta_{2}X_{2ij} + \beta_{3}X_{3ij} + \beta_{4}X_{2ij}X_{3ij} + \color{red}{u_{i}} + \color{red}{v_{j}} + \varepsilon_{ij}$$
where $$y$$ is indexed by $$i$$ for the $$i$$-th subject (ID) and by $$j$$ for the $$j$$-th plot. The random intercepts for ID $$u_{i}$$ and for plot $$v_{j}$$ are colored in red.