> mm <- read.csv("http://www.ats.ucla.edu/stat/data/mmreg.csv")
> colnames(mm) <- c("Control", "Concept", "Motivation", "Read", "Write", "Math", "Science", "Sex")
> psych <- mm[, 1:3]
> acad <- mm[, 4:8]

Using library CCA obtain correlation matrices within and between the two sets of variables

> require(CCA)
> matcor(psych, acad)

Calculate the canonical correlation between the two sets of variables. columns 3 and 4 contain the x and y coefficients

> cc1 <- cc(psych, acad)
> cc1$cor        # display the canonical correlation coefficients
> names(cc1)
> cc1[3:4]        # display the x and y coefficients

Test for significance of the number of dimensions. This is not available in a single package and needs to be calculated. To do this first intialize a couple of the dimensions (with sincere thanks to idre @UCLA)

> n <- dim(psych)[1]
> p <- length(psych)
> q <- length(acad)

then perform the basic calculations

>  {
   ev <- (1 - cc1$cor^2)
   k <- min(p, q)
   m <- n - 3/2 - (p + q)/2
   w <- rev(cumprod(rev(ev)))
   d1 <- d2 <- f <- vector("numeric", k)
   for (i in 1:k) {
       s <- sqrt((p^2 * q^2 - 4)/(p^2 + q^2 - 5))
       si <- 1/s
       d1[i] <- p * q
       d2[i] <- m * s - p * q/2 + 1
       r <- (1 - w[i]^si)/w[i]^si
       f[i] <- r * d2[i]/d1[i]
       p <- p - 1
       q <- q - 1
   }
   pv <- pf(f, d1, d2, lower.tail = FALSE)
}

then perform and report the tests of significance for the canonical axes

> dmat <- cbind(WilksL = w, F = f, df1 = d1, df2 = d2, p = pv)
> dmat

to standardize canonical coefficients for psych SDs

> s1 <- diag(sqrt(diag(cov(psych))))
> s1 %*% cc1$xcoef

to standardize canonical coefficients for acad SDs

> s2 <- diag(sqrt(diag(cov(acad))))
> s2 %*% cc1$ycoef