Is it possible to “auto-complete” an orthogonal contrast matrix to make it square in R and produce correct estimates?

I have a factor with eight levels but have only have five comparisons I want to make. I create a matrix with all of the relevant contrasts, however, I need to invert this matrix prior to passing it to the contrasts() function in order to generate correct estimates. Given that I need a square matrix, is it possible to auto-generate two more orthogonal contrasts so that the Estimates returned by summary() are correct?

http://rstudio-pubs-static.s3.amazonaws.com/65059_586f394d8eb84f84b1baaf56ffb6b47f.html (See sections "DIY Contrasts" and "Running Fewer than J-1 Contrasts for J Groups")

Reproducible example below. Note: the model doesn't make much sense with the CO2 dataset, but the contrasts are exactly the ones I need to make for my original dataset.

# Use CO2 dataset for example
data<-CO2
# Only need eight levels
data<-data[data$Plant %in% levels(data$Plant)[1:8],]
data$Plant<-factor(data$Plant)
levels(data$Plant) # Set up contrasts contrasts(data$Plant) <-
solve( t( cbind(
c(1,1,1,1), # Filler
c(-0.5,0,1,0,-0.5,0,0,0),
c(-0.5,0.5,0,0, -0.5, 0.5, 0, 0),
c(-0.5,0,0,1, -0.5, 0, 0, 0),
c(0,-0.5,0,-0.5, 0, 0.5, 0, 0.5),
c(-0.5,0,-0.5, 0, 0.5, 0, 0.5,0)) )
) [,2:6] #Drop filler

# Run model
fit<-lm(uptake~Plant, data=data)
summary(fit)

Thank you, Egor

contrasts(data$Plant) <- MASS::ginv( t( cbind( c(1,1,1,1), # Filler c(-0.5,0,1,0,-0.5,0,0,0), c(-0.5,0.5,0,0, -0.5, 0.5, 0, 0), c(-0.5,0,0,1, -0.5, 0, 0, 0), c(0,-0.5,0,-0.5, 0, 0.5, 0, 0.5), c(-0.5,0,-0.5, 0, 0.5, 0, 0.5,0)) ) ) [, 2:6] However, you don't need the filler at all if you compute the generalized inverse. I provide some details in my answer here. contrasts(data$Plant) <-